Algebra 2
3.2 Solving Linear
Systems Algebraically
Substitution and Elimination
Learning Targets
• Students should be able to…
• Use algebraic methods to solve linear
systems (both substitution and
elimination).
• Use linear systems to model real-life
situations.
Warm-up
• Solve each equation for the indicated
variable.
1. 2x – y = 5 Solve for y.
2. -x+2y = 3 Solve for x.
3. 3x – 4y = 12 Solve for y.
4. 3x – 4y = 12 Solve for x.
Homework Check
• 3.1
• Page 142
#12, 18, 41, 42, 43, 46, 49
Review of Formulas
• Slope Intercept Form:
y = mx + b
• Standard Form:
Ax + By = C
• Point Slope Form:
Substitution
• Steps to Solving a Linear Equation by Substitution:
• Solve one of the equations for one of the variables
• Substitute this expression into the other equation and
solve for the remaining variable.
• Take this value and substitute it into the equation you
found from step 1 and solve for the remaining
variable.
• Check! (in original equations)
Example 1: Solve the following systems
using the substitution method.
a. 3x – y = 13
2x + 2y = –10
b. x + 3y = –2
–4x – 5y = 8
c. x – y = 8
–3x + 6y = –24
d. 2x – y = 14
6x + 3y = 18
Elimination/Linear Combination
•
Multiply one or both equations by a number so
coefficients for 1 of the variables are opposite.
•
Add the equations eliminating one of the
variables. Solve for the remaining variable.
•
Substitute this value into one of the original
equations and solve for the other variable.
•
CHECK! (in original equations)
Use Elimination to Solve
1. 3x – 5y = –36
6x + 2y = 0
2. 2x + 3y = –1
–5x + 5y = 15
3. 6x + 2y = 20
-4x + 3y = –22
4. 2x – 6y = 19
–3x + 2y = 10
Special Results
1. 6x – 4y = 14
–3x + 2y = 7
2. 4x – 6y = 14
–6x + 9y = –21
Three ways to solve a
system of linear equations:
• 1. Graphing
**Easy to graph – in slope intercept form or
in an easy standard form
• 2. Substitution
**Easy to substitute – one of the variables
has the coefficient of 1 or –1
• 3. Linear Combination (Elimination)
**Easy to linear combine – coefficients in one
equation is a multiple of the coefficient
of the same variable in the other equation.
Real World Problems
1. A citrus fruit company plans to make 13.25 lb gift boxes of
oranges and grapefruits. Each box is to have a retail value of
$21.00. Each orange weighs .5 lb and has a retail value of
$.75, while each grapefruit weighs .75 lb and has a retail
value of $1.25. How many oranges and grapefruits should be
included in the box?
Real World Problems
2. You are planting a 160 sq ft garden with shrubs
and perennial plants. Each shrub costs $42 and
requires 16 sq ft of space. Each perennial plant
costs $6 and requires 8 sq ft of space. You plan to
spend a total $270. How many of each type of
plant should you buy to fill the garden?
Closure
Homework Assignment
• 3.2
• Page 152 – 153
#substitution: 11, 18,
elimination: 24, 34,
any method: 35, 36, 39, 44, 47, 48