2.2 PSAT/SAT Algebra Practice; Linear Equaitons

Which of the following equations is a linear equation?

• y = 2x + 3z + 5

• y - 3x³ + 4x² + 2x + 8

• y = 4x - 6

• y = x² - 21

Answer explanation

Read through the given equations to see how many variables are present. If the equation is between more than 2 variables (i.e. y and x), this equation is not linear.

In choice A, we actually have 3 variables, since there are x, y, and z. We can therefore eliminate choice A as an option to be a linear equation.

Option B has an x to the third power term and an x to the second power term, so we can eliminate B as a potential linear equation.

Option C only has an x to the first power term, so we will keep it as a potential linear equation.

Option D has an x to the second power term, so we can eliminate D as a potential linear equation.

Which of the following equations is a linear equation?

• y = -6x + 5

• y = x³ - 7

• y = x²

• y = 2x² + 4x + 8

Answer explanation

Step 1: Read through the given equations to see how many variables are present. If the equation is between more than 2 variables (i.e. y and x), this equation is not linear.

All equations only have 2 variables, x and y, so we can't remove any equations at this step.

Step 2: Read through the given equations to determine what exponents the variables are raised to. If the variables are raised to any exponent other than one, the equation is not linear.

Choice A only has x raised to the first power, so we will keep it as an option for a linear equation.

Choices B, C, and D can be removed as options for linear equations because they contain terms where x is raised to the second or third power.

Which of the following statements about a LINEAR EQUATION IN TWO VARIABLES is FALSE?

• A linear equation in two variables has a finite number of solutions.

• • The graph of a linear equation in one or two variables always forms a straight line and it can be extended indefinitely in both directions.

• • The value of the variable that makes a linear equation true is called the solution or root of the linear equation.

• Every solution of

  a linear equation can be represented by a unique point on the graph of the equation.

Answer explanation

A linear equation in two variables has infinitely many solutions.

Which of the following statements about LINEAR and NONLINEAR graphs is FALSE?

• The equation determines whether the function is linear or nonlinear. The function will be linear if the highest exponent of x in the equation is one, otherwise it will be nonlinear.

• A linear graph forms a straight line when it is plotted on a graph, while a nonlinear equation is curved in some way.

• It is also possible to see that it is linear by looking at the formula: the formula for a straight line is always

  y = mx + b.

• The slope of a non-linear graph is the same everywhere, whereas the slope of a linear graph changes all the time.

Answer explanation

The slope of a linear graph is the same everywhere, whereas the slope of a non-linear graph changes all the time.

The degree of the equation is the highest power (exponent) of x in the two variable equation. For example, in the equation y = 3x⁷ + x³ + 2 the degree of the equation is 7. To distinguish linear vs nonlinear equations: when the degree of the equation is 1, the equation is linear; otherwise, it is nonlinear. Of the questions below, which of the following is a linear equation?

• y = x² + x + 3

• y = -32.8x + 4z + 3

• y = -1/3x + -3/5

• 3y = 5x³ + 2

Answer explanation

y = x² + x + 3 is incorrect because x is squared.

y = -32.8x + 4z + 3 is incorrect because there are three variables: x, y, and z.

y = -1/3x + -3/5 is correct because this is slope-intercept form of a linear equation: y = mx + b and x has an exponent of 1.

3y = 5x³ + 2 is incorrect because x has an exponent that does not equal 1.