On a long bike ride, Alex pedaled at an average speed of 12 miles per hour for hours on flat terrain and then at 8 miles per hour for hours on a hilly section. If the total distance traveled was 76 miles and , what is the value of ?

Solution available with step-by-step explanation

SAT Linear Equations in Two Variables Question 140 | Free SAT Prep | Aniko

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Question 140·Medium·Linear Equations in Two Variables

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On a long bike ride, Alex pedaled at an average speed of 12 miles per hour for $x$ hours on flat terrain and then at 8 miles per hour for $y$ hours on a hilly section.

If the total distance traveled was 76 miles and $y = 2$ , what is the value of $x$ ?

Your answer

(Express the answer as an integer)

For rate-and-distance word problems, always start by writing distance = rate × time for each segment of the trip. Turn the words into algebraic expressions (like 12x and 8y), then add them to match the given total distance. Once you have a single equation, substitute any known values (such as y = 2) and solve the resulting linear equation carefully, watching your arithmetic when isolating the variable.

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Write expressions for each distance

Use distance = rate × time. What expression represents the distance traveled on the flat part? What expression represents the distance on the hilly part?

Use the total distance

Add the two distance expressions together and set their sum equal to 76 miles, since that is the total distance traveled.

Substitute the known time on hills

You know $y = 2$ . Substitute this into your equation so that only $x$ is left, then solve the resulting one-variable equation.

Desmos Guide

Enter the equation in Desmos

In a new expression line, type 12x + 8*2 = 76. This represents the total distance equation after substituting $y = 2$ .

Find the x-value that satisfies the equation

Desmos will show a solution for $x$ (either as text under the expression or by allowing you to tap the equation). The displayed value of $x$ is the number of hours Alex rode on flat terrain.

Step-by-step Explanation

Translate the word problem into an equation

Use distance = rate × time for each part of the trip.

On flat terrain: rate = 12 miles per hour, time = $x$ hours, so distance on flat = $12x$ miles.
On hilly terrain: rate = 8 miles per hour, time = $y$ hours, so distance on hills = $8y$ miles.

The total distance is the sum of these two distances:

12x + 8y = 76

Substitute the given value of y

You are told that $y = 2$ hours on the hilly section.

Substitute $y = 2$ into the equation:

12x + 8(2) = 76

Simplify $8(2)$ :

12x + 16 = 76

Solve the linear equation for x

Solve $12x + 16 = 76$ step by step.

First subtract 16 from both sides:

\begin{aligned} 12x &= 76 - 16 \\ 12x &= 60 \end{aligned}

Now divide both sides by 12:

x = \frac{60}{12} = 5

So the value of $x$ is 5.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation