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

Number of possible solutions
Systems of linear equations
The heart of algebra

LEARNING OBJECTIVE

After this lesson, you will be able to:

· Determine the number of possible solutions for a system of linear equations, if any

To answer a question like this:

In the system of linear equations above, k represents a constant. What is the value of 3k if the system of linear equations has no solution?

A. 20

B. 30

C. 60

D. 80

You need to know this:

The solution to a system of linear equations consists of the values of the variables that make both equations true.

A system of linear equations may have one solution, infinitely many solutions, or no solution.

If a system of equations represents two lines that intersect, then the system will have exactly one solution (in which the x− and y-values correspond to the point of intersection).

If a system of equations has infinitely many solutions, the two equations actually represent the same line. For example, 2x + y = 15 and 4x + 2y = 30 represent the same line. If you divide the second equation by 2, you arrive at the first equation. Every point along this line is a solution.

If a system of equations has no solution, as in the question above, the lines are parallel: there is no point of intersection.

You need to do this:

· If the question states that the system has no solution, set both x-coefficients equal to each other and both y-coefficients equal to each other to make the lines parallel, but be sure that the y-intercepts (or constant terms, if the equations are in ax + by + c form) are different.

· If the question states that the system has infinitely many solutions, make the x-coefficients equal, the y-coefficients equal, and the y-intercepts (or constant terms) equal.

· If the question states that the system has one solution and provides the point of intersection, substitute the values at that point of intersection for x and y in the equations.

Explanation:

Start by recognizing that for two lines to be parallel, both the x-coefficients must be equal and the y-coefficients must be equal. Manipulate the second equation so that it is in the same format as the first one:

The y-coefficient in the first equation, 10x − 4y = 8, is 4. Divide the second equation by 2 in order to make the y-coefficients in both equations equal:

Now, set the x-coefficient equal to that in the first equation:

Note that the question asks for the value of 3k, so the correct answer is (C), 60.

Try on Your Own

Directions: 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.

HINT: How can the x− and y-values you are given as the solution to the system in Q11 help you find h and k?

11. What is the value of if the (x, y) solution of the above system of equations is (−5, 2)?

A.

B.2

C.

D. 6

HINT: For Q12, if a system of equations has infinitely many solutions, what do you know about the two equations?

12. If q is a constant and the above system of equations has infinitely many solutions, what is the value of q ?

A. −9

B.

C.

D. 9

HINT: For Q13, what does it mean, graphically, when a system has no solution?

13. In the system of linear equations shown above, z is a constant. If the system has no solution, what is the value of z ?

A.

B.

C.8

D. 10

14. For which of the following values of w will the system of equations above have no solution?

A. −8

B.−4

C.4

D. 8

15. If the system of linear equations shown above has infinitely many solutions, and c is a constant, what is the value of c ?

A.

B.

C.2

D. 12