Algebra

Simultaneous equations

Move two lines. Find a shared solution.

Predict, test and explain

Step 1 of 3

Predict

Do y = x + 1 and y = x + 3 have a shared solution?

{y=x+1y=−x+3\begin{cases}y=x + 1\\y=-x + 3\end{cases}
−10-10
−10-10
−8-8
−8-8
−6-6
−6-6
−4-4
−4-4
−2-2
−2-2
22
22
44
44
66
66
88
88
1010
1010
xx
yy

Scroll the diagram sideways to see every label.

First liney=x+1y=x + 1Second liney=−x+3y=-x + 3

The solution satisfies both equations.

InvestigateCan you make two lines with no shared point?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

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Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

Solve these equations together.

y=2x+1,y=−x+7y=2x+1,\qquad y=-x+7

Hint

At the intersection, both expressions give the same output.

Reveal answer

Equate the two expressions, solve for the input, then substitute to find the output.

x=2,y=5x=2,\qquad y=5

Build an explanation

Hint 1

Look for a point that belongs to both lines.

Hint 2

At their intersection, both equations give the same y-value.

Hint 3

Equate the two expressions for y, solve for x, then substitute to find y.

Worked example

This example uses fixed values, separate from the diagram controls.

Solve y = x + 1 and y = −x + 3 together.

  1. Equate the expressions: x + 1 = −x + 3.
  2. Add x and subtract 1: 2x = 2, so x = 1.
  3. Substitute: y = 1 + 1 = 2.

Answer: x = 1 and y = 2.

Connect this idea

Watch out: Different parallel lines have no solution; identical lines share infinitely many solutions.