Systems of Linear Equations
Model two conditions, eliminate efficiently, and interpret the one pair that satisfies both.
Systems of Linear Equations
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A solution must satisfy both equations
A system describes two conditions involving the same unknowns. Its solution is the ordered pair that makes both equations true—the intersection of their graphs.
Substitution
Replace one variable using an equivalent expression.
Elimination
Add or subtract equations so one variable disappears.
Let the equations tell you which method is faster
Problem: Find the pair that satisfies x+y=9 and x−y=1.
Recognize: The y-terms are opposites: +y and −y. They disappear when the equations are added.
Do it fast: Add: 2x=10, so x=5. Substitute into x+y=9: 5+y=9, so y=4.
Answer: (5,4). Check it in both original equations.
Why it works
Elimination is fastest here because one variable disappears immediately. If an equation already said y=…, substitution would usually be faster.
Five forms you should recognize
Substitute the pair into both equations.
x+y=9 and x−y=1 add to 2x=10.
y=2x+1 into x+y=10.
Use one equation for number sold and another for total value.
The intersection satisfies both linear rules.
Check before you commit
- Checking only one equation
- Mixing x- and y-values
- Adding equations without aligning terms
- Modeling money without prices
- Stopping after finding one variable
- Assuming every system has one solution
Do you need the lesson-or just practice?
One original question in each form recommends your next step. It does not yet verify mastery.
Work at the level you need.
Foundations
Build the core procedure with immediate explanations.
Core Practice
Use mixed forms with less scaffolding.
UPCAT-Style Transfer
Apply the competency in unfamiliar representations.
Ready to verify this competency?
A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.
Systems of Linear Equations FAQ
Why are there two mastery sets?
A different five-question set appears after an unsuccessful attempt so memorizing one set does not verify mastery.
Does 5/5 mean I never need this topic again?
It verifies this competency today; revisit it later through mixed practice.
Continue your mathematics review.
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Mastery results save to your Teacher Abi study profile.
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