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 – 5

A. 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

A. f(x) and g(x) are consistent and dependent.

B. f(x) and g(x) are inconsistent.

C. f(x) and h(x) are consistent and independent.

D. g(x) and h(x) are consistent.

D. Graph the system of equations below. Which statement is not true? f(x) = x + 2 g(x) = x + 4

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

Example 6

Read the Test ItemYou are given a system of two linear equations and are asked to find the solution.

No Solution and Infinite Solutions

Solve the system of equations.2x + 3y = 125x – 2y = 11

A. (2, 3)

B. (6, 0)

C. (0, 5.5)

D. (3, 2)

Example 6No 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

Example 6Replace x with 3 and solve for y.

No Solution and Infinite Solutions

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.

Example 6Solve the system of equations.x + 3y = 72x + 5y = 10

A.

B. (1, 2)

C. (–5, 4)

D. no solution

Homework

P. 141 # 3 – 11 odd, 19 – 25 odd, 31 – 41 odd, 51 – 57 odd