PSAT/NMSQT Prep 2022 - Eggert M.D., Strelka A. 2022
Substitution
Systems of linear equations
The heart of algebra
LEARNING OBJECTIVE
After this lesson, you will be able to:
· Solve systems of linear equations by substitution
To answer a question like this:
What is the value of y if 5x + 3y = 20 and x + y = 20?
A. −40
B. −20
C. 20
D. 40
You need to know this:
A system of two linear equations simply refers to the equations of two lines. “Solving” a system of two linear equations usually means finding the point where the two lines intersect. (However, see the lesson titled “Number of Possible Solutions” later in this chapter for exceptions.)
There are two ways to solve a system of linear equations: substitution and combination. For some PSAT questions, substitution is faster; for others, combination is faster. We’ll cover combination in the next lesson.
You need to do this:
To solve a system of two linear equations by substitution, do the following:
· Isolate a variable (ideally, one whose coefficient is 1) in one of the equations.
· Substitute the result into the other equation.
Explanation:
Isolate x in the second equation, then substitute the result into the first equation:
Thus, (D) is correct. If you needed to know the value of x as well, you could now substitute 40 for y into either equation to find that x = −20.
Try on Your Own
Directions: Solve these questions using substitution. Take as much time as you need on these questions. Work carefully and methodically. There will be an opportunity for timed practice at the end of the chapter.
1. Clarice had twice as many nickels as dimes in her piggy bank. When she adds 4 more nickels, she has three times as many nickels as dimes. What was the total number of nickels and dimes in Clarice’s piggy bank before she added the additional nickels?
A. 4
B.8
C.12
D. 16
HINT: Ask yourself: Which variable in Q2 is the easier one to isolate?
2. What is the value of b that satisfies 5c + 5b = 20 and 5b − c = 4?
3. What is the value of x − y from the solution of the above system of equations?
A. −5
B.0
C.2
D. 5
HINT: Since the correct answer to Q4 requires you to know the value for r, solve for and substitute s in terms of r.
4. If 3r + 2s = 24 and r + s = 12, what is the value of r + 6?
A. 0
B.4
C.6
D. 12
5. At a certain restaurant, there are 25 tables and each table has either 2 or 4 chairs. If a total of 86 chairs accompany the 25 tables, how many tables have exactly 4 chairs?
A. 7
B.12
C.15
D. 18