Another representation commonly used for linear relationships is an equation, or symbolic representation.

Source: Matthewrutledge, Balcones-canyonlands-nat-wildlife-refuge.jpg, Wikimedia

The Rimrock Trail in Balcones Canyonlands National Wildlife Refuge in central Texas begins at an elevation of 1,100 feet at the base along the creek bed and ascends to an elevation of 1,300 feet at the top of the canyon.

The Hernandez family used the following equation to describe their elevation above sea level, y, as a function of time since the hike began, x.

The Hernandez family also enjoys hiking in other places. Match each equation with the description of the change in slope, or rate of change, by dragging the description to the space next to the equation.

Pause and Reflect

How can you determine the changes in the slope from an equation in slope-intercept form, y = mx + b?

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Check Your Answer

The slope is the value of m, which is the coefficient of x. Any changes in slope or rate of change relate to comparisons of the value of m between two or more equations.

Practice

1. Belinda and her cousin, Cynthia, were each driving from their separate homes to visit their grandparents. Belinda used the equation y = 120 − 65x, where x represents the time (in hours) since she left home, to determine the number of miles, y, remaining until reaching her grandparents’ home. Cynthia used the equation y = 180 – 55x to determine the number of miles remaining until she reached her grandparents’ home. Which cousin drove faster toward their grandparents’ home?

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   Need a hint?

   Compare the slopes of both equations.
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   Check Your Answer

   Belinda, who drove at a rate of 65 miles per hour, is faster than Cynthia’s speed of 55 miles per hour.
2. Rhonda borrowed money to purchase a used car. She can use the equation y = 6,000 – 250x, where x represents the number of months she will be paying the loan, to determine y, the remaining loan balance. If Rhonda pays an additional $75 per month, what will be the new equation she can use?

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   Will the $75 be added to the 6,000 or the 250?
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   Check Your Answer

   y = 6,000 – 325x
3. Antoine belongs to a gym that charges an activation fee of $65 and a monthly fee of $18. He can use the equation y = 18x + 65, where x represents the number of months that he has been a member of the gym, to determine y, the total amount he has paid for his membership. If the gym increases the monthly fee by 15%, what will be the new equation that Antoine can use?

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   Need a hint?

   If the monthly fee increases by 15%, how much more will Antoine need to pay per month?
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   Check Your Answer

   y = 20.70x + 65