Problem 34 - Understanding Elementary Algebra with Geometry

Question

Sets of values are given for variables having a linear relationship. In each case, write the slope-intercept form for the equation of the line corresponding to the given set of values and answer the accompanying question. $$\begin{array}{|l|c|c|} \hline x \text { (Number of hours practicing video game) } & 2 & 3 \\ \hline y \text { (Grade on math exam) } & 75 & 70 \\ \hline \end{array}$$ What would the grade be if a student practices video games for 4 hours?

Step-by-Step Solution

Answer

The grade would be 65.

Step 1 - Identify the coordinates

Extract the coordinates from the given data. The first coordinate is \(2, 75\) and the second coordinate is \(3, 70\).

Step 2 - Calculate the slope (m)

Use the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) to find the slope. Substitute the coordinates into the formula: \(m = \frac{70 - 75}{3 - 2} = \frac{-5}{1} = -5\).

Step 3 - Find the y-intercept (b)

Use the slope-intercept form \(y = mx + b\) and one of the coordinates to solve for b. Using the point \(2, 75\) and the calculated slope (m): \(75 = -5(2) + b\). Solve for b: \(75 = -10 + b\), \(b = 85\).

Step 4 - Write the equation of the line

Combine the slope and y-intercept to write the equation in slope-intercept form: \(y = -5x + 85\).

Step 5 - Calculate the grade for 4 hours of practice

Substitute \(x = 4\) into the equation \(y = -5(4) + 85\): \(y = -20 + 85 = 65\).

Step 6 - Conclude the result

When a student practices video games for 4 hours, the grade on the math exam would be 65.

Key Concepts

Slope-Intercept Form

Slope-intercept form is one of the most popular formats for writing the equation of a line. In this form, an equation is written as:

\[ y = mx + b \]

The formula is both handy and informative:

• \