Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The...
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Transcript of Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The...
![Page 1: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/1.jpg)
Solve by Graphing
Solve the system of equations by graphing.x – 2y = 0x + y = 6
The graphs appear to intersect at (4, 2).
Write each equation in slope-intercept form.
![Page 2: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/2.jpg)
Solve by Graphing
Check Substitute the coordinates into each equation.
x – 2y = 0 x + y = 6 Original equations
4 – 2(2) = 0 4 + 2 = 6 Replace x with 4and y with 2.
? ?
0 = 0 6 = 6 Simplify.
Answer: The solution of the system is (4, 2).
![Page 3: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/3.jpg)
Which graph shows the solution to the system of equations below?x + 3y = 7x – y = 3
A. C.
B. D.
![Page 4: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/4.jpg)
Classify Systems
A. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent.x – y = 5x + 2y = –4
Write each equation in slope-intercept form.
![Page 5: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/5.jpg)
Classify Systems
Answer:
The graphs of the equations intersect at (2, –3). Since there is one solution to this system, this system is consistent and independent.
![Page 6: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/6.jpg)
Classify Systems
B. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent.9x – 6y = –66x – 4y = –4
Write each equation in slope-intercept form.
Since the equations are equivalent, their graphs are the same line.
![Page 7: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/7.jpg)
Classify Systems
Answer:
Any ordered pair representing a point on that line will satisfy both equations. So, there are infinitely many solutions. This system is consistent and dependent.
![Page 8: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/8.jpg)
Classify Systems
C. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent.15x – 6y = 05x – 2y = 10
Write each equation in slope-intercept form.
![Page 9: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/9.jpg)
Classify Systems
Answer:
The lines do not intersect. Their graphs are parallel lines. So, there are no solutions that satisfy both equations. This system is inconsistent.
![Page 10: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/10.jpg)
Classify Systems
D. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent.f(x) = –0.5x + 2g(x) = –0.5x + 2h(x) = 0.5x + 2
![Page 11: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/11.jpg)
Classify Systems
Answer:
f(x) and g(x) are consistent and dependent. f(x) and h(x) are consistent and independent. g(x) and h(x) are consistent and independent.
![Page 12: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/12.jpg)
A. Graph the system of equations below. What type of system of equations is shown? x + y = 52x = y – 5A. consistent and independent
B. consistent and dependent
C. consistent
D. none of the above
![Page 13: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/13.jpg)
B. Graph the system of equations below. What type of system of equations is shown? x + y = 32x = –2y + 6
A. consistent and independent
B. consistent and dependent
C. inconsistent
D. none of the above
![Page 14: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/14.jpg)
C. Graph the system of equations below. What type of system of equations is shown?
y = 3x + 2–6x + 2y = 10A. consistent and independent
B. consistent and dependent
C. inconsistent
D. none of the above
![Page 15: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/15.jpg)
![Page 16: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/16.jpg)
![Page 17: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/17.jpg)
Use the Substitution Method
FURNITURE Lancaster Woodworkers Furniture Store builds two types of wooden outdoor chairs. A rocking chair sells for $265 and an Adirondack chair with footstool sells for $320. The books show that last month, the business earned $13,930 for the 48 outdoor chairs sold. How many of each chair were sold?
Understand
You are asked to find the number of each type of chair sold.
![Page 18: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/18.jpg)
Use the Substitution Method
Define variables and write the system of equations. Let x represent the number of rocking chairs sold and y represent the number of Adirondack chairs sold.
x + y = 48 The total number of chairs sold was 48.
265x + 320y = 13,930 The total amount earned was $13,930.
Plan
![Page 19: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/19.jpg)
Use the Substitution Method
Solve one of the equations for one of the variables in terms of the other. Since the coefficient of x is 1, solve the first equation for x in terms of y.
x + y = 48 First equation
x = 48 – y Subtract y from each side.
![Page 20: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/20.jpg)
Use the Substitution Method
Solve Substitute 48 – y for x in the second equation.
265x + 320y = 13,930 Second equation
265(48 – y) + 320y = 13,930 Substitute 48 – y for x.
12,720 – 265y + 320y = 13,930 Distributive Property
55y = 1210 Simplify.
y = 22 Divide each side by 55.
![Page 21: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/21.jpg)
Use the Substitution Method
Now find the value of x. Substitute the value for y into either equation.
x + y = 48 First equation
x + 22 = 48 Replace y with 22.
x = 26 Subtract 22 from each side.
Answer: They sold 26 rocking chairs and 22 Adirondack chairs.
![Page 22: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/22.jpg)
Use the Substitution Method
Check You can use a graphing calculator to check this solution.
![Page 23: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/23.jpg)
A. 210 adult; 120 children
B. 120 adult; 210 children
C. 300 children; 30 adult
D. 300 children; 30 adult
AMUSEMENT PARKS At Amy’s Amusement Park, tickets sell for $24.50 for adults and $16.50 for children. On Sunday, the amusement park made $6405 from selling 330 tickets. How many of each kind of ticket was sold?
![Page 24: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/24.jpg)
![Page 25: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/25.jpg)
Solve by Using Elimination
Use the elimination method to solve the system of equations.
x + 2y = 10x + y = 6
In each equation, the coefficient of x is 1. If one equation is subtracted from the other, the variable x will be eliminated.
x + 2y = 10
(–)x + y = 6
y = 4 Subtract the equations.
![Page 26: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/26.jpg)
Solve by Using Elimination
Now find x by substituting 4 for y in either original equation.
x + y = 6 Second equation
x + 4 = 6 Replace y with 4.
x = 2 Subtract 4 from each side.
Answer: The solution is (2, 4).
![Page 27: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/27.jpg)
A. (2, –1)
B. (17, –4)
C. (2, 1)
D. no solution
Use the elimination method to solve the system of equations. What is the solution to the system?x + 3y = 5x + 5y = –3
![Page 28: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/28.jpg)
No Solution and Infinite Solutions
Read the Test ItemYou are given a system of two linear equations and are asked to find the solution.
Solve the system of equations.2x + 3y = 125x – 2y = 11
A. (2, 3)
B. (6, 0)
C. (0, 5.5)
D. (3, 2)
![Page 29: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/29.jpg)
No Solution and Infinite Solutions
x = 3
Solve the Test ItemMultiply the first equation by 2 and the second equation by 3. Then add the equations to eliminate the y variable.
2x + 3y = 12 4x + 6y = 24Multiply by 2.
Multiply by 3.
5x – 2y = 11 (+)15x – 6y = 3319x = 57
![Page 30: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/30.jpg)
No Solution and Infinite Solutions
Replace x with 3 and solve for y.
2x + 3y = 12 First equation
2(3) + 3y = 12 Replace x with 3.
6 + 3y = 12 Multiply.
3y = 6 Subtract 6 from each side.
y = 2 Divide each side by 3.
Answer: The solution is (3, 2). The correct answer is D.
![Page 31: Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e2c5503460f94b1a8bc/html5/thumbnails/31.jpg)