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

Systems of quadratic and linear equations
Quadratics
Passport to advanced math

LEARNING OBJECTIVE

After this lesson, you will be able to:

· Solve a system of one quadratic and one linear equation

To answer a question like this:

In the xy-plane, two equations, and , intersect at points (0, 5) and (a, b). What is the value of −b?

You need to know this:

You can solve a system of one quadratic and one linear equation by substitution, exactly as you would for a system of two linear equations. Alternatively, if the question appears on the calculator section, you can plug the system into your graphing calculator.

You need to do this:

· Isolate y in both equations.

· Set the equations equal to each other.

· Put the resulting equation into the form .

· Solve this quadratic by factoring, completing the square, or using the quadratic formula. (You are solving for the x-values at the points of intersection of the original two equations.)

· Plug the x-values you get as solutions into one of the original equations to generate the y-values at the points of intersection. (Usually, the linear equation is easier to work with than the quadratic.)

Explanation:

Start by isolating y in both equations to get and . Now, set the right sides of the equations equal and solve for x:

The question says that (0, 5) is one point of intersection for the two equations and asks for the y-value at the other point of intersection, so plug x = 1 into either of the original equations and solve for y. Using the linear equation will be faster:

So (a, b) = (1, −1). The question asks for −b, so grid in 1.

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: You can take a shortcut in Q26. There’s no need to isolate y in the second equation.