To prove that the sum of the exterior angles of any polygon is equal to 360 degrees, the Sum of interior angles = (n - 2) x 180 degrees.
we can use the following steps:
Draw any polygon and mark a point outside the polygon for each vertex.
Draw a line segment connecting each vertex to the point outside the polygon that corresponds to it, creating an exterior angle at each vertex.
Measure each exterior angle using a protractor and record the measurements.
Sum all the exterior angles together and observe the result.
We can see that the sum of all the exterior angles is always equal to 360 degrees, regardless of the number of sides or the shape of the polygon. This can be written as:
The sum of exterior angles = 360 degrees
To prove this algebraically, we can use the fact that the sum of the interior angles of any polygon is given by the formula:
Sum of interior angles = (n - 2) x 180 degrees
where n is the number of sides of the polygon. Each exterior angle is supplementary to the corresponding interior angle, so we can write:
The sum of exterior angles = Sum of interior angles (supplementary angles)
Substituting the formula for the sum of interior angles, we get:
Sum of exterior angles = (n - 2) x 180 degrees (supplementary angles)
= 180n - 360 degrees
= 360 degrees - 360 degrees + 180n
= 360 degrees - Sum of interior angles
= 360 degrees - (n - 2) x 180 degrees
= 360 degrees - 180n + 360 degrees
= 720 degrees - 180n
Since the sum of the exterior angles must be a positive value, we can take the absolute value and simplify:
|Sum of exterior angles| = |720 degrees - 180n|
= 180|4 - n|
Since n is always a positive integer greater than or equal to 3 (since a polygon must have at least three sides), we know that 4 - n is always a negative integer between -1 and -n+1. Therefore, |4 - n| is equal to n - 4. Substituting this into the equation above, we get:
|Sum of exterior angles| = 180|n - 4|
= 180(n - 4)
= 180n - 720 degrees
= 360 degrees - 360 degrees + 180n - 720 degrees
= 360 degrees - Sum of interior angles
Therefore, we have shown that the sum of the exterior angles of any polygon is equal to 360 degrees.
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A blue car is 600 kg moving at 5m/s crashes into a red car 400 kg which is initially at rest. After the crash the two cars stick together. What is the initial momentum of the red car before the Collison
Suppose that the functions s and t are defined for all real numbers x as follows.
s(x)=3x+1
T(x)=6x
Write the expressions for (t-s)(x) and (t+s)(x) and evaluate (t x s)(4)
to evaluate (t x s)(4), we substitute x = 4 into the expression for (t x s)(x). This gives us (18x + 6)|x=4, which simplifies to 72 + 6 = 78.
(t-s)(x) = 6x - (3x+1) = 3x - 1
(t+s)(x) = 6x + (3x+1) = 9x + 1
(t x s)(4) = (6x)(3x+1)|x=4 = (18x + 6)|x=4 = 72 + 6 = 78
First, we need to define the functions s and t given by s(x)=3x+1 and t(x)=6x. We can then use these functions to calculate (t-s)(x) and (t+s)(x). To calculate (t-s)(x), we subtract s(x) from t(x). This gives us 3x - 1. To calculate (t+s)(x), we add s(x) and t(x). This gives us 9x + 1. Finally, to evaluate (t x s)(4), we substitute x = 4 into the expression for (t x s)(x). This gives us (18x + 6)|x=4, which simplifies to 72 + 6 = 78.
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What is the equation of the line shown in this graph?
The equation of the line shown in the graph, from (–2,2) to (–3,–1) is y = -5/4x – 7/4.
What is equation?Equation is a mathematical statement that states the equality of two expressions. It contains an equal sign (=) and expresses the relationship between two values, expressions or functions. Equations are used to solve for unknown values, to describe relationships between variables, and to model and solve real-world problems.
To solve for the equation of the line, we can use the point-slope formula, y – y1 = m(x – x1), where m is the slope of the line and (x1, y1) is a point on the line. In this case, the point (x1, y1) is (–2,2) and the slope of the line is m = (–1 – 2)/(–3 – (–2)) = –3/–1 = 3. Therefore, the equation of the line is y – 2 = 3(x – (–2)).
Simplifying, we get y = 3x + 2. To put the equation in the slope-intercept form, y = mx + b, we can rewrite the equation as y = 3x + 2 = 3x + 2 – 2 = 3x – 2 + 2 = 3x – 2, which is the equation of the line, y = –5/4x – 7/4.
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The equation of the line shown in the graph is y = -3. This equation states that for every y-coordinate on the graph, the corresponding x-coordinate is -3.
What is equation?Equation is a mathematical statement that states the equality of two expressions. It contains an equal sign (=) and expresses the relationship between two values, expressions or functions. Equations are used to solve for unknown values, to describe relationships between variables, and to model and solve real-world problems.
The equation of the line shown in the graph is y = -3. This equation states that for every y-coordinate on the graph, the corresponding x-coordinate is -3. This line is a horizontal line, which means that all points on the line have the same y-coordinate.
To verify this equation, we can use the two points given on the graph: (-2,-3) and (3,-3). Using the slope-intercept form of a line, we can plug in the points to determine the equation of the line.
The slope-intercept form is y = mx + b, where m is the slope, x is the x-coordinate, and b is the y-intercept. The slope is calculated by finding the change in y over the change in x. In this case, both points have the same y-coordinate, so the slope is 0. Therefore, the equation of the line is y = 0x + b.
Plugging in the coordinates of one of the points, (-2,-3), we get -3 = 0x + b. By solving for b, we get b = -3. Substituting b into the equation, we get y = 0x - 3. This is the equation of the line shown in the graph.
In conclusion, the equation of the line shown in the graph is y = -3. This equation states that for every y-coordinate on the graph, the corresponding x-coordinate is -3. This equation was verified by using the slope-intercept form of a line and plugging in the two points given on the graph.
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X:y=1:5
Which is the correct equation?
A)y=5x
B)y=x/5
C)y=x-4
D)y=x+4
The correct equation for x:y= 1:5 is (a) y = 5x.
In mathematics, an equation is a formula that connects two expressions with the equal sign = to indicate that they are equal. The equation consists of two sentences joined by an equal sign ("="). The expressions on both sides of the equation are called the "left" and "right" sides of the equation. Usually the right side of the equation is assumed to be zero. Assuming this doesn't reduce the width because it can be done by subtracting the right side from both sides.
Compare to a scale on which a weight is placed. When the same weight of something (like grain) is placed in two pans, the two weights balance the scales and are said to be equal.
According to the Question:
Given that:
x:y = 1:5
We can write the equation as:
x/y = 1/5
⇒ y = 5x
Therefore x:y = 1:5 is y=5x
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in a statistical problem, a population consists of a) all items of interest in a sample. b) a subject of interest in a sample. c) all items of interest in a statistical problem. d) a subject of interest in a statistical problem
In a statistical problem, a population consists of all items of interest in a statistical problem.The correct option in this case is option C - all items of interest in a statistical problem.
What is a population in statistics?Population refers to the complete set of observations or items for which inferences are being made in statistical inference. It's the group that researchers are interested in studying, and it's the group to which they hope to extend their conclusions.
In statistics, the population is the entire collection of individuals, objects, or measurements that researchers are interested in studying. Researchers utilize statistical techniques to make inferences about population parameters based on data obtained from samples.
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What is the area of th blue glass will be needed
The area of the blue glass needed to fill each dot on the butterfly wing is 144π square inches.
The distance around each dot on the butterfly wing is given as 24π inches. This means that the circumference of each dot is 24π inches. The circumference of a circle is:
Circumference = 2πr
where "r" is the radius of the circle. So, we can write:
2πr = 24π
Simplifying this equation, we get:
r = 12
Therefore, the radius of each dot is 12 inches.
To find the area of the blue glass needed to fill each dot, we can use the formula for the area of a circle:
[tex]Area = \pi r^2[/tex]
Substituting the value of the radius, we get:
[tex]Area = \pi(12)^2[/tex]
Simplifying this equation, we get:
Area = 144π square inches
Therefore, the area of the blue glass needed to fill each dot on the butterfly wing is 144π square inches.
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The complete question is :
An artist is creating a large butterfly sculpture outside a museum. There is a circular dot on each wing made out of metal ring. The distance around each dot is 24π inches. The artist plans to fill the inside of each dot with blue colored glasses. What is the area of the blue glass will be needed to fill each butterfly dot?
Multiply 4 times (x + 2).
Answer:
4x + 8
Step-by-step explanation:
You multiple 4 by x, and then 4 by 2, and add them together
Nihkil surveyed `20` students at his middle school and `13` of them had at least one sibling. There are `300` students at Nikhil’s middle school. If the rest of the school is consistent with these results, about how many students would have at least one sibling?
About 195 students at Nikhil's middle school would have atleast one sibling.
What is Algebraic expression ?
Algebraic expression can be defined as combination of variables and constants.
We can use proportion to estimate the number of students at Nikhil's middle school who have at least one sibling.
If 13 out of 20 students surveyed have at least one sibling, we can assume that the same proportion applies to the entire school.
So, we can set up a proportion as follows:
13/20 = x/300
where x is the number of students in the entire school who have at least one sibling.
Solving for x, we can cross-multiply and get:
13 * 300 = 20 * x
x = (13 * 300)/20
x = 195
Therefore, about 195 students at Nikhil's middle school would have at least one sibling.
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10. Justify Reasoning Find m/QUR. Justify
your answer.
Value of angle ∠QUR in the given figure is 139°.
Define Vertically opposite angleVertically opposite angles are pairs of angles that are opposite to each other when two straight lines intersect. When two lines intersect, they form four angles, and the vertically opposite angles are the angles that are opposite to each other and share a common vertex.
In the given figure;
∠SUR=41°
∠TUS=90°
To find: ∠QUR∠TUS=∠QUN ( Vertically opposite angle)
∠TUN is a straight line, angle subtended over it makes 180°
∠TUS makes 90° so, ∠SUN will be 90°
∠SUN=∠SUR+∠RUN
90°=41°+∠RUN
∠RUN=90°-41°
∠RUN=49°
∠QUR=∠QUN+∠RUN
∠QUR=90°+49°
∠QUR=139°
Hence, Value of angle ∠QUR in the given figure is 139°
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I NEED HELP ON THIS ASAP!!!!
Answer:a
Step-by-step explanation:
Reynaldo is a landlord who owns several rental homes. One all-wood home in
the suburbs is worth $95,000. Because the renters insure their own
belongings, what is the premium Reynaldo pays for property insurance per
year?
Annual Premium per $100 of coverage
Brick
Steel
Building Contents
Area
rating
City
0.43
0.39
Suburb 0.45 0.52
Rural 0.6
0.69
OA. $978.65
B. $950.00
C. $852.00
OD. $788.50
Mixed
Building Contents Building Contents
0.5
0.54
0.56 0.63
0.71
0.8
0.55
0.72
0.89
0.65
0.74
0.91
Wood
Building
0.66
0.83
1
Contents
0.76
0.85
1.02
Answer:
d
Step-by-step explanation:
95,000÷100 then multiply by $0.83 since it's a wood home in the suburbs.
I need help filling in the blanks
Values of x for blanks 4 , 1 in inequality .
What in mathematics is an inequality number?
A relationship between two values in an algebraic statement that are not equal is shown by an inequality. One of the two variables on the two sides of the inequality can be represented by an inequality sign as greater than, greater than or equal to, less than, or equal to another value.
Despite the equals sign being substituted by an arrowhead, the formula 5x 4 > 2x + 3 resembles an equation. It serves as an illustration of inequity. This shows that the left half, 5x 4, is bigger than the right part, 2x + 3.
-3x/4 < 12/1
x < -16
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snowdon has a height of approximately 1100 metres above the sea
assuming that the temperature decreases by 1 oc for every 100m above sea level, work out the temperature at the summit of snowdon when the sea level temperature is 4oc
Answer:
[tex]-7^\circ C[/tex]
Step-by-step explanation:
Let
x -----> the height in meters
y ----> the temperature in Celsius degree
we know that
The linear equation in slope intercept form is equal to
[tex]y+mx+b[/tex]
where
m is the slope or unit rate
b is the y-intercept or initial value
In this problem we have
The slope or unit rate is equal to
[tex]m=-\dfrac{1}{100} \dfrac{^\circ C}{m}[/tex] ---> is negative because is a decreasing function
The y-intercept or initial value is equal to
[tex]b=4^\circ C[/tex] ----> value of y when the value of x is equal to zero (sea level)
substitute
[tex]y=-\dfrac{1}{100} x+4[/tex]
For x=1,100 m
substitute in the linear equation and solve for y
[tex]y=-\dfrac{1}{100} (1,100)+4[/tex]
[tex]x=-7^\circ C[/tex]
air is pumped into a spherical balloon, so the balloon expands. the volume of a sphere of radius r is . if the radius of the sphere after seconds is centimeters, at what rate is air being pumped in when ?
The rate at which air is being pumped in when the radius of the sphere is 10 cm and increasing at a rate of 2 cm/s is 400π/3 cm³/s.
The volume V of a sphere with radius r is given by V = (4/3)πr³. Taking the derivative of this expression with respect to time t, we get dV/dt = 4πr²(dr/dt). This equation relates the rate of change of volume to the rate of change of the radius. We are given that the radius is increasing at a rate of 2 cm/s, so dr/dt = 2 cm/s.
At the instant when the radius is 10 cm, we can find the rate at which air is being pumped in by substituting r = 10 cm and dr/dt = 2 cm/s into the equation for dV/dt. Thus, we have dV/dt = 4π(10)²(2) = 800π cm³/s.
Therefore, when the radius of the spherical balloon is 10 cm and is increasing at a rate of 2 cm/s, the rate at which air is being pumped in is 400π/3 cubic centimeters per second.
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A shipping company makes cube-shaped boxes. Their basic box measures 1 foot per side. They want to know how to scale the basic box to build new boxes of various volumes.
1. If the company wants a box with a volume of 8 cubic feet, by what scale factor do they need to dilate the box?
2. If they want a box with a volume of 10 cubic feet, approximately what scale factor do they need?
3. The company decides to create a graph to help analyze the
relationship between volume (x) and scale factor (y). Complete
the table, rounding values to the nearest hundredth if needed.
Then, on graph paper, plot the points and connect them with a smooth curve.
the points do not fall on a straight line, but rather curve upward as volume increases. This reflects the fact that the scale factor needed to dilate the basic box increases as the volume of the new box increases.
What is volume?Volume is a physical quantity that measures the amount of space occupied by an object, substance, or material. It is usually measured in cubic units such as cubic meters (m³), cubic centimeters (cm³), cubic feet (ft³), or liters (L).
The volume of an object is determined by its three-dimensional dimensions, namely length, width, and height. For example, the volume of a cube is calculated by multiplying its length, width, and height measurements together.
by the question.
To find the scale factor needed to dilate the basic box to a box with a volume of 8 cubic feet, we can use the formula for the volume of a cube: [tex]V=s^{2}[/tex], where V is the volume and s is the length of one side of the cube.
Thus, we can solve for the new length of one side of the cube, which we'll call x:
[tex]8 = x^{3}[/tex]
Taking the cube root of both sides gives:
x = 2
So, the company would need to dilate the basic box by a scale factor of
2 to create a box with a volume of 8 cubic feet.
To find the approximate scale factor needed to dilate the basic box to a box with a volume of 10 cubic feet, we can use the same formula:
[tex]10 = x^{3}[/tex]
Taking the cube root of both sides gives:
x ≈ 2.154
So, the company would need to dilate the basic box by a scale factor of approximately 2.154 to create a box with a volume of 10 cubic feet.
Using the formula. [tex]V=s^{2}[/tex], we can fill in the table as follows:
Volume (cubic feet) Scale Factor
0 0
1 1
5 1.71
8 2
10 2.15
15 2.47
20 2.71
We can then plot these points on a graph with volume on the x-axis and scale factor on the y-axis and connect them with a smooth curve to show the relationship between the two variables.
[tex]10 = x^{3}[/tex]
Taking the cube root of both sides gives:
x ≈ 2.154
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-2+5 what is the answer
Answer:
3
start from -2 and count up 5
-2,-1, 1, 2, 3
Solve the system of equations 5x+2y=24
2X+5y=24
To solve the system of equations:
5x + 2y = 24
2x + 5y = 24
We can use either elimination or substitution method.
Using elimination method:
Multiplying the first equation by 5 and the second equation by -2, we get:
25x + 10y = 120
-4x - 10y = -48
Adding the two equations together eliminates y:
21x = 72
Solving for x:
x = 72/21 = 8/3
Substituting x back into one of the original equations and solving for y:
5(8/3) + 2y = 24
2y = 24 - 40/3
2y = 12/3
y = 6/2 = 3
Therefore, the solution to the system of equations is:
x = 8/3
y = 3
Which vector spaces have exactly one basis?
The vector space that has exactly one basis is the trivial vector space, which is the vector space containing only the zero vector.
A basis for a vector space is a set of linearly independent vectors that span the entire space. In a non-trivial vector space (a space that contains more than just the zero vector), there are infinitely many possible bases. This is because any set of linearly independent vectors that span the space can be used as a basis.
However, in the trivial vector space, there is only one vector (the zero vector), and it is linearly independent of itself. Therefore, the trivial vector space has exactly one basis, which is the set containing only the zero vector. Any other set of vectors in the trivial vector space is linearly dependent, and therefore cannot be used as a basis.
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A pediatrician was interested in the weights of all 14-year-olds in the state of Texas to determine outliers in the data. She gathered data from a random sample of 2,000 pediatricians in the state and wanted to create an appropriate graphical representation for the data.
Which graphical representation would be best for her data?
Bar graph
Circle graph
Histogram
Stem-and leaf plot
The best graphical representation for the pediatrician's data would be a histogram. A histogram is a type of graph that shows the distribution of continuous data. In this case, the weights of the 14-year-olds would likely be continuous data. A histogram would allow the pediatrician to visualize the distribution of weights and identify any outliers or unusual patterns.
Answer:histogram
Step-by-step explanation:
(PLEASE HELP!!)
Technology required. A regular hexagon is inscribed in a circle of radius 1 inch. What is the area of the shaded region? (Lesson 4-10)
The area of the shaded region in the given shapes is determined as 0.544 in².
What is the area of the shaded region?
The area of the shaded region is calculated by subtracting the area of the inscribed hexagon from the area of the circle.
Let's start by finding the side length of the regular hexagon inscribed in the circle.
The distance from the center of the circle to any vertex of the hexagon is equal to the radius of the circle, which is 1 inch.
In a regular hexagon, all sides are equal in length, so each side of the hexagon is also equal to 1 inch.
The area of the regular hexagon can be found using the formula:
Area of a regular hexagon = (3√3/2) x (side length)²
Substituting the side length of 1 inch, we get:
Area of regular hexagon = (3√3/2) x (1 inch)^2
= (3√3/2) square inches
= 2.598 in²
Now, to find the area of the circle, we can use the formula:
Area of a circle = π x (radius)²
Substituting the radius of 1 inch, we get:
Area of circle = π x (1 inch)²
= π square inches
= 3.142 in²
Area of the shaded region = area of circle - area of hexagon
= 3.142 in² - 2.598 in²
= 0.544 in²
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i need help and i can’t fine the right answer
The length of the secant AC, obtained using the tangent secant theorem is AC = 25 units
What is the tangent secant theorem?The tangent secant theorem indicates that when a secant ABC and a tangent, CD meets at an external point C to a circle, the relationship between segments are; AC × BC = CD².
Segment addition postulate indicates that we get;
AC = AB + BC
Where;
AB = 3·x + 1
BC = 9
Therefore;
AC = (3·x + 1) + 9 = 3·x + 10
AC × BC = (3·x + 10) × 9 = 27·x + 90
AC × BC = 27·x + 90
CD = 15
CD² = 15² = 225
AC × BC = CD²
Therefore; 27·x + 90 = 225
27·x = 225 - 90 = 135
x = 135/27 = 5
x = 5
AC = 3·x + 10
Therefore; AC = 3 × 5 + 10 = 25
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The volume of a 3-D figure with the dimensions of 3 2/3, 1 2/3, and 2 1/3
The volume of the 3-D figure is 385/81 cubic units or approximately 4.753 cubic units (rounded to three decimal places).
To find the volume of the 3-D figure, we need to multiply its three dimensions:
Volume = (3 2/3) x (1 2/3) x (2 1/3)
First, we need to convert the mixed numbers to improper fractions:
3 2/3 = 11/3, 1 2/3 = 5/3, and 2 1/3 = 7/3.
Now, we can multiply:
Volume = (11/3) x (5/3) x (7/3)
Volume = 385/81 cubic units, which is the exact answer.
If we want to approximate the answer, we can use a calculator to get:
Volume ≈ 4.753 cubic units (rounded to three decimal places).
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Problem 2: If the garbage company in a city with a population of
1.3 x 10^5 collects approximately 14,300 pounds of garbage per week,
how much garbage per person will they collect in one year?
the garbage company in the city will collect approximately 5.72 pounds of garbage per person in one year.
To calculate how a lot garbage in step with man or woman the garbage enterprise collects in 12 months, we want to find out how many weeks are in a yr and then divide the overall amount of garbage collected in a 12 months by means of the population of the town.
wide variety of weeks in a year: 52 (assuming there aren't any bounce years)
overall amount of garbage collected in a 12 months: 14,three hundred kilos/week x fifty two weeks = 743,600 kilos/12 months
garbage in keeping with character in line with year: 743,six hundred kilos/12 months ÷ 1.three x 10^5 people = 5.72 kilos/person/yryr
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Assume these triangles are similar. What is the value of x?
Answer: X= 15
Step-by-step explanation:
Which of the following sets of ordered pairs represents a function?
((1, 2), (2.1), (2, 2))
((1.2). (2. 1), (0.2))
c ((0.1), (1.0), (0.2))
((-1, 2), (0, 3), (-1, 4))
2 option is Correct for sets of ordered pairs represents a function
The only set of ordered pairs that represents a function is (b), which can also be written as {(1.2, 2), (2, 1), (0.2, 0)}. To be a function, each input (x-value) must correspond to exactly one output (y-value). In set (a), the input value of 2 is associated with two different output values, so it is not a function. Set (c) has two input values that both correspond to the same output value, so it is not a function. Set (d) has two different input values that both correspond to different output values, so it is not a function.
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The chart shows the number of hours each student spends training for their favorite sport per hundred hours write a proportion and make a prediction for the number of hours Simone will spend training over a 2000 hour periods
Answer:240 hours
Step-by-step explanation:100x20=2000
12/100=x/2000 multiply 12 and 100 by 20.
The points Q(6, 3), R(-4,-3), S(-1,-8), and T(9,-2) form rectangle QRST.
Plot the points then click the "Graph Quadrilateral" button. Then find the area of the
rectangle.
The area of the rectangle is 136 square units.
What are quadrilateral graph ?
A quadrilateral graph is a graphical representation of a quadrilateral, which is a four-sided polygon. It is a coordinate system that uses a Cartesian plane, where the x and y axes are used to plot the four vertices of the quadrilateral. The four vertices of the quadrilateral are represented by four points on the graph, and the sides of the quadrilateral are represented by the line segments connecting the points. By examining the coordinates of the vertices and the lengths of the sides, we can determine the properties of the quadrilateral, such as whether it is a parallelogram, a rectangle, a square, a trapezoid, or a kite. Quadrilateral graphs are useful tools for visualizing and analyzing quadrilaterals in geometry.
To find the area of a rectangle, we need to know the length of two adjacent sides. Let's use the points Q and R to determine the length of the rectangle.
Length of QR = √((x₂- x₁)² + (y₂ - y₁)²)
= √((-4 - 6)²+ (-3 - 3)²)
= √((-10)²+ (-6)²)
= √(100 + 36)
= √(136)
Similarly, we can find the length of ST:
Length of ST = √((x₂ - x₁)²+ (y₂ - y₁)²)
= √((9 - (-1))² + (-2 - (-8))²)
= √((10)² + (6)²)
= √(100 + 36)
= √(136)
Since QR and ST are opposite sides of the rectangle and have equal length, we know that the rectangle is indeed a rectangle. Therefore, the area of the rectangle is simply the product of the length and width:
Area of rectangle = length * width
= √(136) * √(136)
= 136 square units
Hence, the area of the rectangle is 136 square units.
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What is an equation of the line that passes through the points (2,5)and(3,6)?
Answer:
y = x + 3
Step-by-step explanation:
BC is tangent to circle D at point C. The measure of
LBDC is 35º.
What is the measure of ZDBC?
1. 35°
2. 55°
3. 90°
4. 180°
Solve the following simultaneous equations using an algebraic method.
4x - 7y = 15
5x - 2y = 12
Answer:
4x −7y=21
x+2y=99
Explanation
4x - 7y = 15
5x - 2y = 12
First equation times 2:
8x - 14y = 30
Second equation times 7:
35x - 14y = 84
Subtract,
27x = 54
x = 54/27 = 2
2y = 5x - 12 = 10 - 12 = -2
y = -1
Answer: x=2, y = -1
Check:
4(2)-7(-1)=8+7=15, good
5(2) -2(-1) = 10 + 2 = 12, good
ax + by = c
dx + ey = f
has solutions
x = (ce-bf)/(ae-bd), y = (a f-cd)/(ae-bd)
4x - 7y = 15
5x - 2y = 12
has solutions
denom = ae-bd = 4(-2) - (-7)(5) = 27
x = (15(-2) - (-7)(12))/27 = (-30 + 84)/27 = 2
y = (4(12)-15(5))/27 = (48-75)/27 = -27/27 = -1
Answer:
(x, y) = (2, -1)
Step-by-step explanation:
You want to solve this system of equations using an algebraic method:
4x -7y = 155x -2y = 12Cross multiplication methodWriting these equations in general form and making an array of coefficients with the x-coefficient repeated, we get the equations and array ...
4x -7y -15 = 05x -2y -12 = 0[tex]\begin{array}{cccc}4&-7&-15&4\\5&-2&-12&5\end{array}[/tex]
Working from the left calculating determinants of pairs of columns, we have ...
∆ = 4(-2) -5(-7) = 27
∆x = -7(-12) -(-2)(-15) = 54
∆y = -15(5) -(-12)(4) = -27
The variable values are ...
x = ∆x/∆ = 54/27 = 2
y = ∆y/∆ = -27/27 = -1
The solution is (x, y) = (2, -1).
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Additional comment
The graph confirms this solution.
The "cross multiplication method" is useful when the coefficients are such that elimination or substitution involves more arithmetic operations than the minimum. The amount of math involved in calculating the x- and y-values is on par with what would be required for "elimination." The method is substantially equivalent to "Cramer's Rule."