Solving Linear Systems Using Linear
Combinations
There are two methods of solving a system of
equations algebraically:
Elimination (Linear Combinations)
- an equation resulting from the sum of two
equations (or multiples of the equations)
- used when no variable in either equation has a
coefficient of 1 or -1
 Substitution

Linear Combinations
Solving a linear system by linear combinations
(addition only):
1. Write the equations so the like terms are lined up in
columns.
2. Add the equations. Combining like terms will
eliminate one variable. Solve for the remaining
variable.
3. Substitute the value from step 2 into either of the
original equations and solve for the other variable.
4. Check the solution.
Linear Combinations
Solve the system:
4x - 8y = 32
16x + 8y = 48
20x = 80
x=4
4x - 8y = 32
4(4) - 8y = 32
16 - 8y = 32
-8y = 16
y = -2
Answer (4, -2)
Add Like Terms
Divide
Original Equation
Substitute
Multiply
Subtract 16 from both sides
Divide
Check
4x - 8y = 32
4(4) - 8(-2) = 32
16 + 16 = 32
32 = 32
16x + 8y = 48
16(4) + 8(-2) = 48
64 + (-16) = 48
64 - 16 = 48
48 = 48
Guided Practice
Solve the system using linear combinations.
3x + 7y = 84
5x - 7y = 12
Arranging Like Terms in Columns
Arrange the like terms in columns. Then solve the
system.
2x = 12 + 8y
4x + 8y = 24
2x - 8y = 12
Re-write the equation to line up
4x + 8y = 24
like terms
6x = 36
Add like terms
x=6
Divide
Arranging Like Terms in Columns
4x + 8y = 24
4(6) + 8y = 24
24 + 8y = 24
8y = 0
y=0
Answer (6, 0)
Check
2(6) = 12 + 8(0)
12 = 12 = 0
12 = 12
Substitute
Multiply
Subtract 24 from both sides
Divide
4(6) + 8(0) = 24
24 + 0 = 24
24 = 24
Guided Practice
Solve the system using linear combinations.
2x - 3y = 18
6y - 2x = 24
Independent Practice
Solve the system using linear combinations.
1.
12x + 5y = 90
-4x - 5y = -50
2.
3x = 9 + 7y
-5y = 3x + 18
Linear Combinations
Solving a linear system by linear combinations (multiply first):
1. Write the equations so the like terms are lined up in
columns.
2. Multiply one equation (or both) by a multiple so that one
of the variables has opposite coefficients in the two
equations. (Find LCM that is opposites.)
3. Add the equations. Combining like terms will eliminate
one variable. Solve for the remaining variable.
4. Substitute the value from step 2 into either of the
original equations and solve for the other variable.
5. Check the solution.
Linear Combinations
Solve the system:
4x - 5y = 29
7x + 10y = 32
2(4x - 5y = 29)
7x + 10y = 32
8x - 10y = 58
7x + 10y = 32
15x = 90
x=6
Multiply equation by 2; LCM = 10
Write new equations
Add like terms
Divide
Linear Combinations
Solve the system:
4x - 5y = 29
7x + 10y = 32
7(6) + 10y = 32
42 + 10y = 32
10y = -10
y = -1
Solution (6, -1)
Substitute
Multiply
Subtract 42 from both sides
Divide
Check
4x - 5y = 29
4(6) - 5(-1) = 29
24 + 5 = 29
29 = 29
7x + 10y = 32
7(6) + 10(-1) = 32
42 + (-10) = 32
42 - 10 = 32
32 = 32
Guided Practice
Solve the system using linear combinations.
4x + 3y = -1
5x + 4y = 1
Writing and Using a Linear System
You drive 210 miles to a relative's house. It takes you
4 hours. Part of the time you are on a freeway,
where the speed limit is 60 mph. The rest of the
time you are on smaller roads, where the speed
limit is 30 mph. Supposing you drove exactly at the
speed limit the whole way, how much time did you
spend on each type of road? Use linear
combinations to solve.
*Think how distance, rate, and time are related
d = rt
Writing and Using a Linear System
(Freeway Time) + (Road Time) = Total Time
(Freeway Speed)(Time) + (Road Speed)(Time) = Distance
Distance = 210 miles
Freeway Time = x
Road Time = y
Freeway Speed = 60
Road Speed = 30
Writing and Using a Linear System
x+y=4
60x + 30y = 210
-30(x + y = 4)
60x + 30y = 210
-30x - 30y = -120
60x + 30y = 210
30x = 90
x = 3
Write Equations
Multiply to by LCM of a variable
Add the two equation
Divide
Writing and Using a Linear System
x+y=4
x = 3
3+y=4
y = 1
Choose Equation
Substitute
Subtract 3 from both sides
Answer You spent 3 hours on the freeway and 1 hour on
the road.
Check your answer.
Independent Practice
Solve the system using linear combinations.
1.
-2x + y = 2
4x - 4y = 4
2.
3y = 8 - 7x
8x - 18 = y