Systems of Linear Equations UPCAT Reviewer: Lesson and Practice

TEACHER ABI UPCAT MATHEMATICS

Systems of Linear Equations

Model two conditions, eliminate efficiently, and interpret the one pair that satisfies both.

5-10 minute lesson27 original questionsAdaptive practiceSaves progress

LessonShortcutExamplesSkill CheckPracticeMasteryFAQ

MY STATUS

Systems of Linear Equations

Not yet verified on this browser.

NOT STARTED

QUICK REVIEW

A solution must satisfy two conditions at the same time

A system of linear equations uses the same unknowns in two conditions. A unique solution is the ordered pair that makes both equations true. Algebraically, solve by substitution or elimination. Graphically, the solution is the intersection.

One intersection → one solutionParallel distinct lines → no solutionThe same line twice → infinitely many solutions

Substitution

Replace one variable using an equivalent expression.

Elimination

Add or subtract equations so one variable disappears.

DO IT FAST

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.

WORKED EXAMPLES

Five forms you should recognize

1. Elimination

x+y=13x−y=1

Add the equations: 2x=14, so x=7. Substitute to obtain y=6.

2. Substitution

y=3x−2x+y=14

Replace y: x+3x−2=14, so x=4 and y=10.

3. Ticket model

For 120 student and adult tickets totaling ₱7,200:

s+a=12050s+80a=7200

The first equation counts tickets; the second counts pesos.

4. No or infinite solutions

Equal slopes with different intercepts are parallel and have no solution. Equivalent equations describe the same line and have infinitely many solutions.

5. Break-even point

For costs 500+3c and 200+5c:

500+3c=200+5c → c=150

At 150 copies the costs match; beyond that, the lower per-copy rate is cheaper.

COMMON TRAPS

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

FIVE-FORM SKILL CHECK

Do you need the lesson-or just practice?

One original question in each form recommends your next step. It does not yet verify mastery.

CHOOSE YOUR PRACTICE

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.

FRESH MASTERY CHECK

Ready to verify this competency?

A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.

QUICK ANSWERS

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.

RELATED COMPETENCIES

Continue your mathematics review.

Linear EquationsLinear FunctionsCoordinate Geometry

SAVE AND CONTINUE

Your progress stays on this browser.

Mastery results save to your Teacher Abi study profile.

Return to Student Hub View UPCAT Coverage