Linear Relationships Examples

Which of these equations represents a linear relationship?

a. y = x + 7
b. y = (x + 2)(x – 4)
c. x = y³
d. y = x + z²

A linear equation has two variables and neither has an exponent. Them's the rules.

Answers a and b both appear to have no exponents, but when answer b is distributed, it ends up having an x² term. The answer is a.

We need to buy some grass skirts for a luau and each one costs $2.00 (tax included). Write a table, a graph, and an equation to represent this relationship, then determine if it's a proportional relationship.

Hula time. If we buy 1 skirt, it'll cost $2.00. If we buy 2 skirts, it'll cost $4.00. If we buy 3 skirts, it'll cost $6.00. We can put this in a table.

To graph the line, we can assume the number of skirts is the x variable and the cost is the y variable. Then we graph it.

To write an equation, we need the slope of the line and the y-intercept. Use the slope formula and the last two points, (2, 4) and (3, 6).

Boom, the slope is m = 2. The line crosses the y-axis at 0, so the y-intercept is b = 0. Now plug 'em both into our slope-intercept equation.

y = mx + b
y = 2x + 0
y = 2x

This relationship is proportional because the equation is in the form y = kx where k = 2. (In other words, it has a slope and no intercept.) We can also see that it's proportional because when we divide any y-value by its x-value, we always get the same number, 2, which is the constant of proportionality.

Write this equation in standard form.

y = 3x + 7

To write this equation in standard form, we move the x-term to the left side of the equal sign so that both the x-and y-terms are together. Trust us, they're happier that way.

Subtract 3x from both sides.

y – 3x = 3x + 7 – 3x
y – 3x = 7

Beautiful. One standard-form equation, served up hot with fries on the side.