Worked example
Example 1: Eliminate a variable cleanly
Problem
What ordered pair solves and ?
Add equations: , so .
Which elimination or substitution step preserves both equations?
Which ordered-pair check satisfies both original equations?
Unit 3 · Linear RelationshipsA4 · Systems · Structures
Systems of Equations
Solve two constraints together and recognize one, zero, or infinitely many solutions.
Ready when you are
7 steps · about 29 minutes
Start with the warm-up.
StartStep 1 of 7Warm-upDone
Warm-up
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Key terms
- System
- Two or more equations required to hold at the same time.
- Intersection
- A point lying on both graphs.
- Elimination
- Combining equations so one variable cancels.
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Step 2 of 7Try it firstDone
Try it first
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Take a shot before the lesson. Write one relationship from the stem, even if you do not finish.
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Step 3 of 7LearnDone
The big idea
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See the structure
A system asks where two conditions are true at once. Substitution replaces one variable; elimination combines equations to cancel one variable.
Key idea
Name quantities before calculating
Write what each number measures. A correct calculation with the wrong quantity or unit still misses the question.
Build a reliable method
On a graph, one intersection means one solution. Parallel distinct lines mean no solution; the same line written two ways means infinitely many.
Explore in Desmos
Read a system intersection
Click the intersection to read . Substitute the pair into both equations: and .
The calculator is live: change anything and see what happens.
Read a system intersection
Degree modePress Escape to leave the calculator (twice if a menu is open).
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Stretch to a 36
For context systems, define what each variable counts before writing equations, then reject solutions that violate the context.
Trap
A believable answer to the wrong question
The choices often include a correct intermediate number. Before selecting it, reread the highlighted ask and check the unit, sign, and domain.
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Worked examples
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Skip check · 2 questions
Already know this? Get both right on the first try and you can skip the worked examples.
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Then , so .
Worked example
Example 2: Use a baseline to model ticket counts
Problem
Adult tickets cost , student tickets ; 30 tickets earn . How many are adult?
If all 30 were student tickets, revenue would be .
Which elimination or substitution step preserves both equations?
Which ordered-pair check satisfies both original equations?
Faded example · you finish it
Example 3: Complete the setup and result
Problem
What ordered pair solves and ?
Choose the setup that follows the stated relationship.
Choose the resulting value.
Desmos way
Solve and check a system graphically
Graph
x+y=11 and 2x-y=7. Click their only intersection and read .Check both equations, not just one: and . Algebraic elimination reaches the same pair by adding the equations to get .
Solve and check a system graphically
Degree modePress Escape to leave the calculator (twice if a menu is open).
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Step 5 of 7PracticeDone
Practice
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Step 6 of 7Spot the mistakeDone
Spot the mistake
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Spot the mistake
Where did it go wrong?
One line has an error. Click the line where it first goes wrong.
What went wrongThe same pair must satisfy both equations. Adding gives .
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Exit ticket
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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36
These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.