Grade 8 · Connect slope and a linear rule · 430 of 750
Connect slope and a linear rule · practice 49
For the line y = -3.5x + (-7), find y when x = 3 and identify the slope and vertical intercept. Follow the calculation, then test the quantities in the animated example.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
For the line y = -3.5x + (-7), find y when x = 3 and identify the slope and vertical intercept.
Use this scratch space or work on paper. Your notes stay on this page and clear when you leave. Answers are for self-checking; they are not automatically graded.
A starting point
Multiply the input by the slope before adding the intercept.
Work through the reasoning
Step 1
Translate the givens
Compare y = -3.5x + (-7) with y = mx + q: m = -3.5, q = -7.
Step 2
Calculate with the model
Substitute x = 3: y = -3.5 × 3 + (-7) = -10.5 + (-7) = -17.5.
Step 3
Check and interpret
At x = 0, y = -7; increasing x from 3 to 4 changes y by -3.5. The intercept and rate of change play distinct roles.
The answer
y(3) = -17.5; slope = -3.5; intercept = -7.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
Use Play, Pause, and the timeline to inspect the construction. Reset restores the question’s original settings. The displayed assumptions describe where this model applies.
Watch the relationship
PausedHD animation studio
From experiment to screen.
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The mathematical idea
In y=mx+b, the slope describes output change per input unit and the intercept gives the output at zero. Changing b translates a graph while leaving its rate unchanged. A negative slope means output decreases as input increases; zero slope gives a constant function. Starting quantities: Slope m = -3.5; Intercept b = -7.
y = mx + b; Δy/Δx = m
03 · Reflect and transfer
Explain what changes and why.
Can two different lines share the same value at x = 3? Explain.
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.