Depth · one concept from first principles · step 2 of 4
Trade a vertical shift for a horizontal one
Shift the line y=3x+1 right 2 and down 1. That same final line can also be reached by a single sideways shift of m units. Find m.
Functions
Your answer
How to solve
Strategy Convert to Algebra — Write each translated line as an equation (shifting by p along x replaces x with x-p; shifting by q along y adds q). Both lines keep slope 3, so setting their y-intercepts equal gives one equation for m.
1STEP 1
Translate the first way
Moving 2 along the x-axis replaces x with x-2, and moving -1 along the y-axis subtracts 1 from the output.
y=3(x-2)+1-1=3x-6
The new line has slope 3 and y-intercept -6.
8.F.A.3Convert To Algebra2STEP 2
Translate the second way
Moving m along the x-axis only replaces x with x-m.
y=3(x-m)+1=3x-3m+1
Same slope 3; the y-intercept is now -3m+1.
8.F.A.3Convert To Algebra3STEP 3
Match the intercepts
Two lines with the same slope coincide exactly when their y-intercepts are equal.
-3m+1=-6 → 3m=7 → m=7/3
One linear equation in m finishes it.
8.F.B.4Organize Information In More WaysThe two translated lines are the same line exactly when their -intercepts are equal.
Why?
Sliding a line moves it without turning it, so both translated lines keep the slope .
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
Why?
Moving the graph to the right means the new point above is the old point that was above , which is why is replaced with .
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
Lines with the same slope are parallel and keep the same gap everywhere, so if they share the point on the -axis the gap is zero everywhere.
🧱Parallel lines keep a constant gapTwo parallel lines never meet and the distance between them is the same everywhere you measure it.
Answer
m=7/3
Check: 3(x-7/3)+1=3x-7+1=3x-6, the same line as the first translation.
Takeaway
On a slanted line, an up-or-down move can always be traded for a sideways move -- the slope sets the exchange rate.
- Translate the first way
- Translate the second way
- Match the intercepts
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested
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