example 2 solve by graphing solve the system of equations by graphing. x – 2y = 0 x + y = 6 the...

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.

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).

Which graph shows the solution to the system of equations below?x + 3y = 7x – y = 3

A. C.

B. D.

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.

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.

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.

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.

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.

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.

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

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.

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

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

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

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.

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

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.

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.

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.

Use the Substitution Method

Check You can use a graphing calculator to check this solution.

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?

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.

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).

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

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)

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

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.