3.2 Solving Systems of Equations Algebraically

Substitution MethodElimination Method

Substitution Method

Here you replace one variable with an expression.

x + 4y = 26

x – 5y = - 10

Solve for a variable,

x = 26 – 4y

Replace “x” in the other equation

(26 – 4y) – 5y = -10

Solve for y

Solve for y

(26 – 4y) – 5y = -10

26 – 4y – 5y = - 10 Remove parentheses by multiplying by 1

26 – 9y = - 10 Add like terms

-9y = - 36 Subtract 26 from both sides

y = 4 Divide by - 9

Solve for x

x = 26 – 4y

x = 26 – 4(4) Substitute for y

x = 26 – 16

x = 10

The order pair is (10, 4). This is where the lines cross.

The Elimination Method

Here we add the equations together when the coefficients are different signs.

x + 2y = 10

x + y = 6

Here both lead coefficients are 1.

We can change the coefficient to – 1, by multiplying by – 1.

x + 2y = 10x + y = 6

Multiply the bottom equation by – 1.

x + 2y = 10- x - y = - 6 When adding the

y = 4 equations together, x go to zero.

Find x by replace it back in either equation.x + 2(4) = 10; x + 8 = 10; x = 2

So the order pair (2, 4)

works in both equations.

2 + 2(4) = 10

2 + 4 = 6

We have two way to solve the systems,

Substitution and Elimination; which way is better depends on the problem.

What about this problem

2x + 3y = 125x – 2y = 11

Here we have to multiply both equationsIf we wanted to remove the “x”, then we

have to find the Least common multiple (L.C.M.) of 2 and 5.

If we wanted to remove the “y”, then we have to find the least common multiple of 3 and -2.

Lets get rid of the “y”

The L.C.M of 2 and 3 is 6. Since we want the coefficients to be opposite, - 2 will help in the equation.

we multiply the top equation by 2.

2x + 3y = 12 4x + 6y = 24

The bottom equation by 3

5x – 2y = 11 15x – 6y = 33

Add the new equations together

4x + 6y = 24 15x – 6y = 3319x = 57Divide by 19

x = 3Replace in original equation and solve for y2(3) + 3y = 12 6 + 3y = 12 3y = 6

y= 2

What about inconsistent systems?

y – x = 5 Multiply the top equation by – 2,

2y – 2x = 8

2y – 2x = -10

then add the bottom. 2y – 2x = 8

0 = - 2

This shows no solutions.

What if it is dependent (Many solutions)

1.6y = 0.4x + 10.4y = 0.1x + 0.25Multiply the top and bottom equation by 100

to remove decimals.160y = 40x + 10040y = 10x + 25Then multiply the bottom equation by -4-160y = -40x – 100

Add the new equations together

160y = 40x + 100

-160y = -40x – 100

0 = 0

This is a system with many solutions.

Solve this system

a – b = 2

-2a + 3b = 3

How about this system

y = 3x – 4

y = 4 + x

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