Combination - Systems of linear equations - The heart of algebra

PSAT/NMSQT Prep 2022 - Eggert M.D., Strelka A. 2022

Combination
Systems of linear equations
The heart of algebra

LEARNING OBJECTIVE

After this lesson, you will be able to:

· Solve systems of linear equations by combination

To answer a question like this:

image

If the lines represented by the equations above intersect at the point (x, y), then what is the value of y?

A. −3

B. −2

C. 2

D. 3

You need to know this:

Combining two equations means adding or subtracting them, usually with the goal of either eliminating one of the variables or solving for a combination of variables (e.g., 5n + 5m).

You need to do this:

To solve a system of two linear equations by combination, do the following:

· Make sure that the coefficients for one variable have the same absolute value. (If they don’t, multiply one equation by an appropriate constant. Sometimes, you’ll have to multiply both equations by constants.)

· Either add or subtract the equations to eliminate one variable.

· Solve for the remaining variable, then substitute its value into either equation to solve for the remaining variable.

Explanation:

Both variables have different coefficients in the two equations, but you can convert the 3x in the second equation to 6x by multiplying the entire second equation by 2:

image

Now that the coefficients for one variable are the same, subtract the second equation from the first to eliminate the x variable. (Note that if the x-coefficients were 6 and −6, you would add the equations instead of subtracting.)

image

Solve this equation for y:

image

(A) is the correct answer. If the question asked for x instead of y, you would now substitute −3 into either of the original equations and solve for x. (For the record, x = 1.)

Try on Your Own

Directions: Solve these questions using combination. 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.

image

6. What is the y-coordinate of the solution to the system of equations shown?

A. −1

B.0

C.image

D. 5

HINT: There’s no need to solve for b and c separately in Q7.

7. If −8c − 3b = 11 and 6b + 6c = 4, what is the value of 3b − 2c ?

A. −27

B.−3

C.8

D. 15

8. If 6a + 6b = 30 and 3a + 2b = 14, then what are the values of a and b ?

A. a = 2; b = 2

B.a = 4; b = 1

C.a = 1; b = 4

D. a = 3; b = 1

9. image Given 2x + 5y = 49 and 5x + 3y = 94, what is the product of x and y ?

image

10. image Sixty people attended a concert. Children’s tickets sold for $8 each and adult tickets sold for $12 each. If $624 was collected in ticket money, how many more adults than children attended the concert?

A. 0

B.12

C.24

D. 60