Systems of equations: one solution shared by two conditions

Where we are on the map

We move from one function to comparing two relationships.

Begin with a familiar image

If two ticket types total 10 and their weighted total is 16, the two conditions determine the counts together.

Precise meaning

A system solution is an ordered pair that makes every equation true. On a graph it is a common intersection point.

The lesson’s support signal

meaning → model → calculation → check → explain in your own words

two conditions → shared variables → eliminate one variable → pair → check both equations

Worked example

Subtracting the first equation from the second gives x=6, then y=4; both original equations check.

Example with a fading prompt

Change the numbers in the example and repeat the solution without copying its steps. Estimate first, calculate exactly, and check against the original condition.

Recall without a prompt

1. Define the main idea in one sentence.
2. Rebuild the support signal from memory.
3. Solve the example with different numbers and explain why every step is valid.

Find and correct the mistake

The pair (5,5) is not a solution merely because it satisfies the first equation; it gives 15 in the second.

Transfer to a new setting

Apply the same relationship to something you can measure yourself. Name the quantities, units, and the boundary within which the model is valid. Check the answer by a second representation or an inverse operation.

Exercise

Required. Solve x+y=14, 3x+y=26 by elimination and graphing, then check both equations.

With your own data. Create one everyday example with the same structure and show it in words, a formula, and a check.

Optional. Seven days later, change the numbers and repeat without the support map.

Open perspective

Powers compress repeated multiplication, and roots ask the inverse question.

Course contents