Ch. 8.2-3: Solving Systems of Equations Algebraically

What is a System of Linear Equations?

If the system of linear equations is going to have a solution, then the solution will be an ordered pair (x , y) where x and y make both equations true at the same time.

We will only be dealing with systems of two equations using two variables, x and y.

We will be working with the graphs of linear systems and how to find their solutions graphically.

A system of linear equations is simply two or more linear equations using the same variables.

inconsistentconsistent

dependentindependent

Definitions Consistent: has at least one solution Dependent: has an infinite amount of

solutions

Inconsistent: has no solutions

Independent: has exactly one solution Consistent: has at least one solution

x

yConsider the following system:

x – y = –1

x + 2y = 5Using the graph to the right, we can see that any of these ordered pairs will make the first equation true since they lie on the line.

We can also see that any of these points will make the second equation true.

However, there is ONE coordinate that makes both true at the same time…

(1 , 2)

The point where they intersect makes both equations true at the same time.

How to Use Graphs to Solve Linear Systems

x – y = –1

x + 2y = 5

How to Use Graphs to Solve Linear Systems

x

yConsider the following system:

(1 , 2)

We must ALWAYS verify that your coordinates actually satisfy both equations.

To do this, we substitute the coordinate (1 , 2) into both equations.

x – y = –1

(1) – (2) = –1 Since (1 , 2) makes both equations true, then (1 , 2) is the solution to the system of linear equations.

x + 2y = 5

(1) + 2(2) =

1 + 4 = 5

Graphing to Solve a Linear System

While there are many different ways to graph these equations, we will be using the slope – intercept form.

To put the equations in slope intercept form, we must solve both equations for y.

Start with 3x + 6y = 15

Subtracting 3x from both sides yields

6y = –3x + 15

Dividing everything by 6 gives us…51

2 2y x=- +

Similarly, we can add 2x to both sides and then divide everything by 3 in the second equation to get

23 1y x= -

Now, we must graph these two equations

Solve the following system by graphing:

3x + 6y = 15

–2x + 3y = –3

Graphing to Solve a Linear System

512 2

23 1

y x

y x

=- +

= -

Solve the following system by graphing:

3x + 6y = 15

–2x + 3y = –3

Using the slope intercept forms of these equations, we can graph them carefully on graph paper.

x

y

Start at the y – intercept, then use the slope.Label the solution!

(3 , 1)

Lastly, we need to verify our solution is correct, by substituting (3 , 1).

Since and , then our solution is correct!( ) ( )3 3 6 1 15+ = ( ) ( )2 3 3 1 3- + =-

Graphing to Solve a Linear System

Let's summarize! There are 4 steps to solving a linear system using a graph.

Step 1: Put both equations in slope – intercept form

Step 2: Graph both equations on the same coordinate plane

Step 3: Estimate where the graphs intersect.

Step 4: Check to make sure your solution makes both equations true.

Solve both equations for y, so that each equation looks like

y = mx + b.

Use the slope and y – intercept for each equation in step 1. Be sure to use a ruler and graph paper!

This is the solution! LABEL the solution!

Substitute the x and y values into both equations to verify the point is a solution to both equations.

SubstitutionSubstitution1. Given two equations. Solve one

equation for a variable.2. Plug this expression in for the variable

into the other equation.3. Solve. 4. Plug this value into the first equation

and solve it as well. 5. Write the answer as an ordered pair.

Example: Solve 1. Solve for p2. Plug into other

equation and solve for q.

3. Plug value of q into initial equation.

4. Write the ordered pair (ABC order)

2 3 2

3 17

p q

p q

3 17p q 17 3p q

2 3 2p q 2( 17 3 ) 3 2q q

34 6 3 2q q 9 36q

4q 3 17p q 3(4) 17p

5p

: ( , )Answer p q( 5,4)

Example: Solve 1. Solve for n2. Plug into other

equation and solve for m.

3. Plug value of m into initial equation.

4. Write the ordered pair (ABC order)

2 10

2

m n

m n

2 10m n 10 2n m

2m n (10 2 ) 2m m 10 2 2m m

3 12m 4m

2 10m n 2(4) 10n

2n

: ( , )Answer m n(4,2)

Elimination1. Given two equations. Make sure variables

are lined vertically2. Choose a variable to eliminate. It must

become the opposite value of the same variable in the other equation.

3. Multiply the entire equation to create the needed values.

4. Add the two equations together5. Solve for variable left.6. Plug value into initial equation and solve.7. Write the answer as an ordered pair

Example: Solve 1. Eliminate n since they

are opposites.2. Add the two equations3. Solve for m4. Plug m back into the

initial equation5. Solve for n6. Write the ordered pair

(ABC order)

2 10

2

m n

m n

2 10

2

m n

m n

3 12m 4m

2 10m n 2(4) 10n

2n : ( , )Answer m n

(4,2)

Example: Solve 1. Eliminate h since they

are opposites. 2. Multiply the second

equation by 2.3. Add the two equations4. Solve for g5. Plug g back into the

initial equation6. Solve for h7. Write the ordered pair

(ABC order)

3 2 10

4 6

g h

g h

11 22g 2g

3 2 10g h 3(2) 2 10h

2 4h

: ( , )Answer g h

(2, 2)

3 2 10

4 6

g h

g h

3 2 10

2(4 6)

g h

g h

3 2 10

8 2 12

g h

g h

2h

Ex#1 Solve the system by substitution. Check by graphing

12 xy

2 2x y – x + 2(-2x + 1) = 2

– x – 4x + 2 = 2

-5x + 2 = 2

-2 -2

-5x = 0

-5 -5

x = 0

y = -2x + 1

y = -2(0) + 1

y = 0 + 1

y = 1

(0, 1)

(x, y)

(0, 1)

12 xy

2 2x y

2 13

4 3 11

x y

x y

Ex#2 Solve the system by substitution.

4x – 3(-2x + 13) = 11

4x + 6x – 39 = 11

10x – 39 = 11

+39 +39

10x = 50

10 10

x = 5

y = -2x + 13

y = -2(5) + 13

y = -10 + 13

y = 3

(5, 3)

(x, y)

2 13y x

Try These: Solve the system by substitution.

2 7x y 943 yx

3(2y – 7) + 4y = 9

6y – 21 + 4y = 9

10y – 21 = 9

+21 +21

10y = 30

10 10

y = 3

x = 2y – 7

x = 2(3) – 7

x = 6 – 7

x = -1

(-1, 3)

(x, y)

2 7x y

6 3 15

2 5

x y

x y

Ex#2 Solve the system by substitution.

6x – 3(2x – 5) = 15

6x – 6x + 15 = 15

15 = 15

Infinite solutions

2 5y x

3 1y x Ex#2 Solve the system by substitution.

6x + 2(-3x + 1) = 5

6x – 6x + 2 = 5

2 5

No Solution

3 1

6 2 5

x y

x y

To eliminate a variable line up the variables with x first, then y. Afterward make one of the variables opposite the other. You might have to multiply one or both equations to do this.

Example #3: Find the solution to the system using elimination.

2 6x – 2y = 16 x + 2y = 5

7x + 0 = 21

7x = 21

7 7

x = 33 + 2y = 5

-3 -3

2y = 2

y = 1 (x, y)

(3, 1)

3x – y = 8

x + 2y = 5

Example #3: Find the solution to the system using elimination.

-3 -6x – 3y = –18 4x + 3y = 24

–2x = 6

x = – 3

2(–3) + y = 6

-6 + y = 6

y = 12

(x, y)

(–3, 12)

2x + y = 6

4x + 3y = 24

3 3x – 4y = 16

5x + 6y = 14

9x – 12y = 48

2

19x = 76

x = 4

(x, y)

(4, -1)

5(4) + 6y = 14

-20 -20

6y = -6

y = -1

20 + 6y = 14

10x + 12y = 28

Example #3: Find the solution to the system using elimination.

Example #3: Find the solution to the system using elimination. 5 -3x + 2y = -10

5x + 3y = 4

-15x + 10y = -50

3

19y = -38

y = -2

(x, y)

(2, -2)

5x + 3(-2) = 4

5x = 10

x = 2

5x – 6 = 4

15x + 9y = 12

Example #3: Find the solution to the system using elimination.12x – 3y = -9

-4x + y = 3

12x – 3y = -9

3

0 = 0

Infinite solutions

-12x + 3y = 9

Example #3: Find the solution to the system using elimination.6x + 15y = -12

-2x – 5y = 9

6x + 15y = -12

3

0 = 15

No solution

-6x – 15y = 27