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Algebra / LESSON 03

Systems of equations · SAT Math

10 min + practice · Original foundation lesson

A solution of a system satisfies both equations simultaneously. For two lines it is their intersection, if one exists.

A reliable method

Add or subtract equations to eliminate a variable, or isolate a variable and substitute. Check the ordered pair in both equations.

The common trap

One matching coefficient does not imply parallel lines. Compare both variable coefficients and then the constants.

Make the connection

A system with no intersection has no solution; coincident lines have infinitely many solutions.

Choose an elimination target

For x+y=11 and 2x−y=4, addition removes y: 3x=15, so x=5 and y=6. Substitution checks both equations: 5+6=11 and 10−6=4. Solving only one equation gives infinitely many candidate pairs; the second condition selects their common pair.

Compare three outcomes

The equations x+y=11 and 2x+2y=22 describe the same line and infinitely many pairs. Replacing 22 with 25 makes the conditions inconsistent. With 2x−y=4 instead, the lines cross once. Compare both variable coefficients and the constant before deciding whether an elimination result such as 0=3 means no solution.

Try, then check

Eight tickets cost $31. Student tickets cost $3 and adult tickets $5. How many adult tickets were sold? An all-student baseline costs $24, leaving $7, which is not divisible by the $2 upgrade. There is no whole-number solution: a realistic answer must satisfy both algebra and the ticket-count constraint. Changing revenue to $32 gives four adult tickets.

WORKED EXAMPLE

If x + y = 9 and x − y = 1, what is x?

  1. Add the equations: 2x = 10.
  2. Divide by 2: x = 5.
  3. Then y = 4; substitution verifies both equations.
Answer: 5
Compare the other answer choices
1

Used the difference x−y as x.

9

Used the total x+y as x.

10

Added equations but forgot to divide by 2.

Try it, then explain it.

Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.

Practice this skill

Sources & authorship

Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.

Official mathematics skillsSiyavula Grade 10 mathematics