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4Solving Systems of Equations and Inequalities

S E C T I O N 4.1

Solving Systems of Equations by Graphing

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Objectives

1. Determine whether a given ordered pair is a solution of

a system.

2. Solve systems of linear equations by graphing.

3. Use graphing to identify inconsistent systems and

dependent equations.

4. Identify the number of solutions of a linear system

without graphing.

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Use graphing to identify inconsistent systems and dependent equations

There are three possible outcomes when we solve a system of two linear equations using the graphing method.

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Determine whether a given orderedpair is a solution of a system

1

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Determine whether a given ordered pair is a solution of a system

Liner equations: with infinitely many pairs of numbers whose sum is 3, there are infinitely many pairs (x, y) that satisfy this equation.

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System of equations:

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Because the ordered pair (2, 1) satisfies both of these

equations, it is called a solution of the system.

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Example 1

Determine whether (–2, 5) is a solution of each system ofequations.

a. b.

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Example 1(a) – Solution

Recall that in an ordered pair, the first number is the x-coordinate and the second number is the y-coordinate.

To determine whether (–2, 5) is a solution, we substitute –2 for x and 5 for y in each equation.

Check:3x + 2y = 4

3(–2) + 2(5) 4–6 + 10 4

4 = 4

The first equation.

True

1111

Example 1(a) – Solution

x – y = –7

–2 – 5 –7

–7 = –7

Since (–2, 5) satisfies both equations, it is a solution of the system.

The second equation.

cont’d

True

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Example 1(b) – Solution

Again, we substitute –2 for x and 5 for y in each equation.

Check:

4y = 18 – x

4(5) 18 –(–2)

20 18 + 2

20 = 20

The first equation.

True

cont’d

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Example 1(b) – Solution

y = 2x

5 2(–2)

5 = –4

Although (–2, 5) satisfies the first equation, it does not satisfy the second.

Because it does not satisfy both equations, (–2, 5) is not a solution of the system.

The second equation.

False

cont’d

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Solve systems of linearequations by graphing

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Solve systems of linear equations by graphing

To use the graphing method to solve

we graph both equations on one set of coordinate axes using the intercept method, as shown below.

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Although there are infinitely many pairs (x, y) thatsatisfy x + y = 3, and infinitely many pairs (x, y) that

satisfy 3x – y = 1, only the coordinates of the point where

their graphs intersect satisfy both equations

simultaneously.

Thus, the solution of the system is (1, 2).

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To check this result, we substitute 1 for x and 2 for y in each equation and verify that the pair (1, 2) satisfies each equation.

Check: First equation Second equation

x + y = 3 3x – y = 1 1 + 2 3 3(1) – 2 1

3 = 3 3 – 2 11 = 1

When the graphs of two equations in a system are different lines, the equations are called independent equations.

True

True

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Example 2

Solve the system of equations by graphing:

2020

Example 2 – Solution cont’d

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From the graph, the solution appears to be (4, –2).

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To check, we substitute 4 for x and –2 for y in each equation and verify that the pair (4, –2) satisfies each equation.

Check:

The equations in this system are independent equations, and the system is a consistent system of equations.

3x = 2y + 16

3(4) 2(–2) + 1612 –4 + 1612 = 12

2x + 3y = 2

2(4) + 3(–2) 28 – 6 2

2 = 2


This is the first equation.

This is thesecond equation.

True True

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Example 4

Solve the system of equations by graphing:

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Example 4 – Solution

Since y = –2x – 6 is written in slope–intercept form, we can graph it by plotting the y-intercept (0, –6) and then drawing a slope of –2. (The rise is –2, and the run is 1.)

y = –2x – 6

So m = –2 =

and b = –6.

We graph 4x + 2y = 8 using the intercept method.

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The system is graphed below. Since the lines in the figure are parallel, they have the same slope.

cont’d

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We can verify this by writing the second equation in slope–intercept form and observing that the coefficients of x in each equation are equal and the y-intercepts are different, (0, –6) and (0, 4).

y = –2x – 6 4x + 2y = 82y = –4x + 8

y = –2x + 4

Because parallel lines do not intersect, this system has no solution and is inconsistent. Since the graphs are different lines, the equations of the system are independent.

cont’d

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Sometimes a system of equations has no solution. Suchsystems are called inconsistent systems.

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There are three possible outcomes when we solve a system of two linear equations using the graphing method.

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Identify the number of solutions of a linear system without graphing

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Identify the number of solutions of a linear system without graphing

We can determine the number of solutions that a system of two linear equations has by writing each equation in slope–intercept form.

• If the lines have different slopes, they intersect, and thesystem has one solution.

• If the lines have the same slope and differenty-intercepts, they are parallel, and the system has nosolution.

• If the lines have the same slope and same y-intercept,they are the same line, and the system has infinitelymany solutions.

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Example 6

Without graphing, determine the number of solutions of:

Strategy:We will write both equations in slope–intercept form.

Solution:To write each equation in slope–intercept form, we solve for y.

5x + y = 5 3x + 2y = 8The first equation. The second equation.

3333


Since the slopes are different, the lines are neither parallel nor identical.

Therefore, they will intersect at one point and the system has one solution.

Different slopes

cont’d

equations and inequalities solving systems of 4 · pdf file10/2/2010 1 solving systems of 4...

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