solving systems of linear equations wait a minute! what’s a system of linear equations? –a...

Solving Systems of Linear Equations

Wait a minute!

• What’s a system of linear equations?– A system is a set of linear sentences which

together describe a single situation.

• How do I know if I have a solution?– A solution to a system is a pair of numbers

that satisfies all equations within the solution.

Examples of Linear Systems

x y 22x y 8

y 9xy 2x 7

The system at the upper left shows an example of two linear equations in standard form (Ax+By=C).

The system at the lower left shows an example of two linear equations in slope-intercept form (y=mx+b).

Non-examples of Linear Systems

y 1

2x 2

y 12x 1

x y 3

The situation in the upper left is a system of equations, but it is not a system of linear equations because the top equation is a quadratic.

The situation at the lower left is not a system because it is not a set of linear sentences. It is only one linear sentence.

Which of the following is a system of linear equations?

y x2x 3y 15

y 2x 3

y 1x

3

*Note: The equations are brought in as objects from Equation Editor. Therefore, press the button in the blue space below the equations.

Be Careful…

• This is a system of equations, but both equations are not linear.

• Linear equations always have x raised to the first power.

1

xx 1

Review Examples of Linear Systems

Great Job!

Now that you can identify a system of linear equations, it is time to learn how to solve them!

Solving Systems of Linear Equations

Solving by Graphing

Solving by Substitution

Solving by Elimination

Steps to Solve by Graphing

1. Graph both equations in the system.2. Find the ordered pair at the point of

intersection.3. This ordered pair is the solution to the

system! (It contains the x and y values that will make both equations true!) Always check your solution in the equations.

Important Note:

It is VERY important to graph your lines accurately! Use graph paper and a straight edge!

Let’s look at an example!

Solve

63

12

xy

xy

First, graph y=2x+1. Remember, plot the y-intercept first (1) and then use the slope (2) to find another point on the line.

Solve

63

12

xy

xy

Second, graph y=-3x+6 on the same set of axes.

The solution is the point of intersection!

The lines intersect at (1,3). This means that when x=1, then y=3 in BOTH equations in the system. To be sure, our next step is to check the solution in the equations.

Does (1,3) work in both equations? Let’s check!

33

123

1123

12

xy

33

633

6133

63

xy

(1,3) is the solution to

63

12

xy

xy

Since (1,3) is the point of intersection and it worked in both equations of the system, this is the solution!

Test your understanding.

Solve

1532 yx

xy

The first step is to graph both of these equations.

Solve

1532 yx

xy

Choose the graph below that has both equations graphed correctly. (Click in the yellow area below the graph!)

Now check your solution!

11 xy

151

1532

1513)1(2

1532

yx

The intersection point is (-1,1). Does this work in both equations?

11 xy

The solution does not work in either equation! This means the graph is not correct!

Now check your solution!

22 xy

1510

1564

15)2(3)2(2

1532

yx

The intersection point is (-2,-2). Does this work in both equations?

The solution does not work in the second equation! This means the graph is not correct!

Now check your solution!

1515

1596

15)3(3)3(2

1532

yx

33 xy

The intersection point is (-3,-3). Does this work in both equations?

Great job! (-3,-3) works in both equations so it is the solution!

Solving by Substitution

Look at the system

278

255

xy

xy

278255 xyx

We can say that

Solving by Graphing

• You have done a great job solving systems of equations by graphing.

• There are two other techniques to solving systems of linear equations. – Solving by Substitution– Solving by Elimination

• Return to the Home Menu to learn more!

By the Transitive Property…

• The Transitive Property states that if a=b and b=c, then a=c.

• From our system, we know that 5x-25=y and y=-8x+27.

• Therefore, 5x-25=-8x+27!– This is an equation with one variable. We

can solve for x!

Solve the Equation

278255 xxx8 x8

272513 x25 25

5213 x13 13

4x

X=4 is part of the solution!

• When x=4, both equations will result in the same y value. This is the other coordinate in our solution!

• Substitute x=4 into one of the equations. (It does not matter which equation you use; both will give the same result for y.)

Let’s find the y value!

5

2520

2545

255

y

y

y

xy

5

2732

2748

278

y

y

y

xy

Regardless of which equation you use, when x=4 then y=-5. Therefore, the solution to this system is (4,-5).

Rules for Solving by Substitution

1. Solve both equations for y.

2. Set the equations equal to each other.

3. Solve for x.

4. Substitute the x value into one of the equations to solve for y.

5. Once you have x and y, write your solution as an ordered pair.

Try this example.

Using substitution, solve

104

3

52

1

xy

xy

Set the equations equal to each other and solve for x. When you do this, what result do you get?

X=3 X=5 X=12

If x=3, solve for y.

5.32

72

10

2

3

52

3

532

1

52

1

y

y

y

y

y

xy

Does the solution (3,-3.5) check in both equations?

Check the solution (3,-3.5)

5.35.3

55.15.3

532

15.3

52

1

xy

75.75.34

315.3

4

40

4

95.3

104

95.3

1034

35.3

104

3

xy(3,-3.5) does not work in the second equation. Therefore, this is not the solution.

If x=5, solve for y.

5.2

55.2

552

1

52

1

y

y

y

xy

Does the solution (5,-2.5) check in both equations?

Check the solution (5,-2.5)

5.25.2

55.25.2

552

15.2

52

1

xy

25.65.24

255.2

4

40

4

155.2

104

155.2

1054

35.2

104

3

xy(5,-2.5) does not work in the second equation. Therefore, this is not the solution.

If x=12, solve for y.

1

56

5122

1

52

1

y

y

y

xyDoes the solution (12,1) check in both equations?

Check the solution (12,1)

11

561

5122

11

52

1

xy

11

1091

10124

31

104

3

xyGreat job! Since (12,1) works in both equations, it is the solution to this system!

Solving by Substitution

• You have done a great job solving systems of equations by substitution.

• There are two other techniques to solving systems of linear equations. – Solving by Graphing– Solving by Elimination

• Return to the Home Menu to learn more!

Solving by Elimination

• We know that if a=b and c=d, then a+c=b+d.– Apply this to equations to solve by

elimination.

• The goal is to add the equations of a system to get a variable to cancel out.

Solve

150

120

yx

yx

150

120

yx

yx

2702 x

First, add this equations together. Since there is –y in the first equation and +y in the second equation the y variable will cancel out!

2 2135x

Now that we know what x is, we can solve for y!

If x=135, solve for y!

150135

150

y

yx

13515y

You can use either equation to solve for y.

135

Our solution is (135,15). To verify that this is the solution, we can check the coordinates in both equations.

Check the solution (135,15)

120120

12015135

120

yx

150150

15015135

150

yx

Since (135,15) works in both equations, it is the solution to this linear system! Now try an elimination problem on your own…

Solve

642

282

yx

yx

When you add the equations in the system together, which variable will you end up solving for?

x y

Check again…

• When you add the equations, 2x+-2x will cancel out, leaving you y to solve for.

Good job!

Now solve for y! What is the result?

Y=2/3 Y=1 Y=2

150

120

yx

yx

If y=2/3, solve for x.

3

53

102

3

6

3

162

23

162

23

282

282

x

x

x

x

x

yx

Does the solution check in both equations?

3

2,3

5

Check the solution

3

2,3

5

22

23

6

23

16

3

10

23

28

3

52

282

yx

63

2

63

8

3

10

63

24

3

52

642

yx

This ordered pair does not work in the second equation. Therefore, it is not the solution to the system.

If y=1, solve for x.

3

62

282

2182

282

x

x

x

x

yxDoes the solution (-3,1) check in both equations?

Check the solution (-3,1)

22

286

21832

282

yx

62

646

614)3(2

642

yx

This ordered pair does not work in the second equation. Therefore, it is not the solution to the system.

If y=2, solve for x.

7

142

2162

2282

282

x

x

x

x

yxDoes the solution (-7,2) check in both equations?

Check the solution (-7,2)

22

21614

22872

282

yx

66

6814

624)7(2

642

yx

Good job! This ordered pair works in both equations. Therefore, (-7,2) is the solution to the system!

Solving by Elimination

• You have done a great job solving systems of equations by elimination.

• There are two other techniques to solving systems of linear equations. – Solving by Graphing– Solving by Substitution

• Return to the Home Menu to learn more!• If you have already learned all three methods,

click on the blue arrow for more information!

Solving Systems of Equations

You have now learned the basic principles to solving systems of linear equations using the three methods: graphing, substitution, and elimination.

For more practice on solving systems of linear equations (or to look at more advanced examples) click here.

Back to the beginning… End