Math  /  Algebra

QuestionGreg is driving to Miami. Let yy represent his distance from Miami ( related by the equation 42565x=y425-65 x=y.
Answer the questions below. Note that a change can be an increase or a decrease. For an increase, use a positive number. For a decrease, use a nega
What was Greg's distance from Miami when he began his drive? \square miles What is the change in Greg's distance from Miami for each hour he drives \square miles

Studdy Solution

STEP 1

What is this asking? How far was Greg from Miami at the **start** of his drive, and how much closer does he get to Miami for every hour he drives? Watch out! The equation shows how Greg's distance from Miami *decreases* as he drives, so make sure the change in distance is negative!

STEP 2

1. Initial Distance
2. Change in Distance

STEP 3

Alright, so Greg's distance from Miami is given by this cool equation: y=42565xy = 425 - 65x.
Here, yy is the distance, and xx is the number of hours he's been driving.
We want to find his **initial distance**, which means we're looking for his distance at the very beginning of his trip.

STEP 4

At the start of his drive, Greg hasn't driven *any* time at all!
So, the number of hours he's driven, xx, is **zero**: x=0x = 0.
Let's plug that into our equation!

STEP 5

y=425650y = 425 - 65 \cdot 0 y=4250y = 425 - 0 y=425y = 425

STEP 6

So, when x=0x = 0, we find that y=425y = 425.
This means Greg's **initial distance** from Miami was 425\textbf{425} miles!

STEP 7

Now, we want to figure out how Greg's distance from Miami *changes* every hour he drives.
We can see from the equation y=42565xy = 425 - 65x that for every hour (xx) Greg drives, his distance (yy) from Miami decreases.

STEP 8

The equation is already in a super helpful form called **slope-intercept form**: y=mx+by = mx + b.
Here, mm is the **rate of change**, or how much yy changes for every increase in xx, and bb is the **initial value** of yy when x=0x = 0.

STEP 9

Our equation is y=42565xy = 425 - 65x, which is the same as y=65x+425y = -65x + 425.
So, our **rate of change** is m=65m = -65.

STEP 10

This means that for every hour Greg drives, his distance from Miami changes by -65\textbf{-65} miles.
The negative sign tells us that his distance *decreases*, which makes sense because he's driving *towards* Miami!

STEP 11

Greg's initial distance from Miami was **425** miles.
The change in Greg's distance from Miami for each hour he drives is **-65** miles.

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