Download Solving Systems using Substitution

LESSON 3: USING A SUBSTITUTION STRATEGY TO SOLVE A
SYSTEM OF LINEAR EQUATIONS
Learning Outcome: Learn to use the substitution of one variable to solve a
linear system.
Work with a partner.
Solve each linear system without graphing:
 3x + 5y = 6
x = -4
 2x + y = 5
y = -x + 3
What strategies did you use?
How can you check that each solution is correct?
In the previous lessons, we solved linear systems by graphing. This
strategy is time-consuming even when you use graphing technology, and
you can only approximate the solution. We use algebra to determine an
exact solution. One algebraic strategy is called solving by substitution.
By using substitution, we transform a system of two linear equations into a
single equation in one variable, then we use what we know about solving
linear equations to determine the value of that variable.
The skill of substituting algebraic expressions is used regularly in math and
science. The substitution method can provide a quick solution to a linear
system.
Solve the following system:
4x + 5y = 26
3x = y – 9
Ex. Solve this linear system:
5x – 3y = 18
4x – 6y = 18
Ex. Admission to the Abbotsford International Airshow cost $80 for a car
with two adults and three children. Admission for a car with two adults cost
$50. Use algebra to determine the cost for one child and the cost for one
adult. There was no charge for the vehicle or parking.
These two linear systems have the same graph and the same solution
x = 1 and y = 2.
System A
𝑥
2
1
3
+𝑦 =
System B
5
Multiply each term by 2
2
1
1 Multiply each term by 3
3
3
𝑥− 𝑦=−
𝑥 + 2𝑦 = 5
𝑥 − 𝑦 = −1
In system A, we can multiply the first equation by 2, and the second by 3 to
write equivalent equations with integer coefficients. The result is the
equations in system B.
The two linear systems have the same solution (1, 2) because the
corresponding equations are equivalent. Multiplying or dividing the
equations in a linear system by a non-zero number does not change the
graphs. So, their point of intersection, and hence, the solution of the linear
system is unchanged.
A system of equivalent equations is called an equivalent linear system
and has the same solution as the original system.
When an equation in a linear system has coefficients or a constant term
that are fractions, we can multiply by a common denominator to write an
equivalent equation with integer coefficients.
Ex. Solve this linear system by substitution.
1
2
4
𝑥 − 𝑦 = −2
5
1
3
4
8
𝑦= 𝑥−
Assignment: pg. 424-427 #1-22, 27