EST I HS Math · Unit 1 · Foundational Algebra
Lesson 1.5 · Systems of Equations
On the paper: 1–3 questions per real paper; 'no solution / infinitely many' appears almost every year
Big idea. Two equations are two lines. They cross once, never, or everywhere. The EST rarely wants the crossing point itself — it wants a combination of x and y, or the value of a letter that makes the lines parallel.
1Elimination: line the equations up and add ➕
🍼 Baby steps
- 1
x + 2y = −1 2x = y + 8
December 2024. First, write both in the same shape: 2x − y = 8.
- 2
x + 2y = −1 and 4x − 2y = 16
Double the second so the y-terms are opposites (+2y and −2y).
- 3
add: 5x = 15 → x = 3
The y disappears. That is elimination. The question asked for x, so stop here.
- 4
substitution instead: y = 2x − 8 → x + 2(2x − 8) = −1 → 5x = 15
Same answer; use substitution when one equation already has x or y alone.
🎛 Play with it · two lines, three outcomes
Slide the coefficients. Make the lines parallel (a₁/a₂ = b₁/b₂) and the solution vanishes; match c as well and they become one line.
1x + 2y = -1
2x − 1y = 8
One solution: the lines cross at (3, -2).
ratio test: a₁/a₂ = 0.5 · b₁/b₂ = -2 · c₁/c₂ = -0.13
a₁/a₂ = b₁/b₂ ⇒ parallel (no solution) — unless c₁/c₂ matches too, then it is one line (infinitely many). Different ratios ⇒ exactly one solution.
2Do not solve — combine 🧮
🍼 Baby steps
- 1
13x − 7y = 12 7x − 13y = 6
December 2020 asks for 4x + 4y. Solving for x and y separately is a minute of fractions.
- 2
subtract: 6x + 6y = 6
(13 − 7)x + (−7 + 13)y = 12 − 6. Look what appeared: a multiple of x + y.
- 3
x + y = 1 → 4x + 4y = 4
Twelve seconds. Whenever the question asks for a symmetric combination (x + y, x − y), add or subtract first.
- 4
−2x + 5y = 39, 3x + 4y = 45 → 2x + 7y = ?
May 2021. Here no shortcut lands exactly, so solve: y = 9, x = 3, then 6 + 63 = 69. Check the shortcut first; fall back to solving.
3No solution / infinitely many: the ratio test ∥
🍼 Baby steps
- 1
a₁x + b₁y = c₁ a₂x + b₂y = c₂
Two lines. Their slopes are −a₁/b₁ and −a₂/b₂.
- 2
a₁/a₂ = b₁/b₂ ≠ c₁/c₂ ⇒ no solution (parallel)
Same slope, different line.
- 3
a₁/a₂ = b₁/b₂ = c₁/c₂ ⇒ infinitely many (same line)
One equation is a multiple of the other.
- 4
2x + 15y = 18, kx − 5y = −7, no solution → k/2 = −5/15 → k = −2/3
June 2021. One proportion, no solving. (Check c: −7/18 ≠ −1/3, so it is genuinely parallel.)
- 5
ax − ½y = c, 2x + 4y = 5, infinitely many → a/2 = (−½)/4 = c/5
December 2020. (−½)/4 = −1/8, so c = 5 × (−1/8) = −5/8.
Trap · the sign of the ratio
4Word problems that are systems 📝
🍼 Baby steps
- 1
20 tables seat 88 people; some seat 4, some seat 5
February 2022. Two unknowns (x four-seat tables, y five-seat) → two equations.
- 2
x + y = 20 4x + 5y = 88
One equation counts tables, the other counts seats.
- 3
4(x + y) = 80 → 88 − 80 = 8 = y
Trick: if every table seated 4 there would be 80 seats; the 8 extra seats are one per five-seat table. y = 8, x = 12.
5Real EST questions, solved twice 📄
x + 2y = −1 2x = y + 8
The value of x that satisfies the system of equations above is:
- A.1
Would need y = −1 from the first equation, but then 2x = 7.
- B.2
Then y = −1.5 and 2(2) ≠ −1.5 + 8.
- C.3
- D.4
y would be −2.5 and 8 ≠ 5.5.
Safe way · ~40 s
- From the second: y = 2x − 8.
- x + 2(2x − 8) = −1 → 5x = 15 → x = 3.
⏱ 90-second way · ~15 s
- Test the options in both equations: x = 3 → y = −2 from the first; 2(3) = −2 + 8 ✓.
Answer: C
−2x + 5y = 39 3x = −4y + 45
From the system of equations above, what is the value of 2x + 7y?
- A.−57
Sign error on x (x = −3 instead of 3).
- B.15
Only 2x + y.
- C.39
The right-hand side of the first equation, copied.
- D.69
Safe way · ~60 s
- Second: 3x + 4y = 45. Multiply first by 3, second by 2: −6x + 15y = 117 and 6x + 8y = 90.
- Add: 23y = 207 → y = 9. Then 3x + 36 = 45 → x = 3.
- 2(3) + 7(9) = 6 + 63 = 69.
⏱ 90-second way · ~20 s
- Add the two equations as written: x + 9y = 84. Not the target — so solve. y = 9, x = 3 → 69.
- Calculator EQN with (−2, 5, 39) and (3, 4, 45) gives x = 3, y = 9 in 15 s.
Answer: D
2x + 15y = 18 kx − 5y = −7
What is the value of k if the system of simultaneous equations above admits no solutions?
- A.−1/3
That is b₁/b₂ inverted (−5/15). k must equal 2 × (−1/3).
- B.−6
Multiplied 2 by 3 instead of by −1/3.
- C.−2/3
- D.2
Sign dropped and the ratio inverted.
Safe way · ~45 s
- Slopes: −2/15 and k/5. Parallel: −2/15 = k/5 → k = −2/3.
- Check intercepts differ: 18/15 ≠ −7/(−5) = 1.4 ✓ (1.2 ≠ 1.4).
⏱ 90-second way · ~12 s
- Ratio test: k/2 = −5/15 = −1/3 → k = −2/3.
Answer: C
ax − ½y = c 2x + 4y = 5
The system of equations above has infinitely many solutions. If a and c are constants, what is the value of c?
- A.−5/8
- B.1/4
That is −a (a = −1/4), not c.
- C.5/4
Sign lost and c divided by the wrong number.
- D.11
A guess from adding coefficients.
Safe way · ~50 s
- Same line ⇒ all three ratios equal: a/2 = (−½)/4 = c/5.
- (−½)/4 = −1/8. So c = 5 × (−1/8) = −5/8 (and a = −1/4).
⏱ 90-second way · ~15 s
- Scale the second equation by −1/8 (to turn 4y into −½y): −¼x − ½y = −5/8. Read c = −5/8.
Answer: A
13x − 7y = 12 7x − 13y = 6
Based on the system of equations above, what is the value of 4x + 4y?
- A.4
- B.6
6x + 6y = 6 — that is the intermediate line, not 4x + 4y.
- C.18
Added the right-hand sides (12 + 6).
- D.24
4 × 6, mixing the two above.
Safe way · ~90 s
- Solve fully: from elimination x = 0.9, y = 0.1 (fractions all the way).
- 4(0.9) + 4(0.1) = 4.
⏱ 90-second way · ~12 s
- Subtract: 6x + 6y = 6 → x + y = 1 → 4x + 4y = 4. The question was built for this.
Answer: A
⏱ The 90-second lane
One question = 90 seconds. These are the moves that finish this topic in one line.
- 1Asked for x + y or x − y? Add or subtract the equations first — the combination often falls out in one line.
- 2Ratio test: a₁/a₂ = b₁/b₂ ⇒ parallel; also = c₁/c₂ ⇒ same line. Keep the signs.
- 3"No solution" with an unknown k: set k/a₂ = b₁/b₂ — one proportion, no solving.
- 4Substitute when a variable is already alone (y = 2x − 8); eliminate otherwise.
- 5Options are numbers? Test them in BOTH equations — one equation is never enough.
- 6Word problems: one equation counts items, the other counts money/seats/points.
- 7Calculator: MODE → EQN → 2 unknowns gives x and y in about 15 s when the coefficients are ugly.
Take with you
- ▸Two lines: one crossing, parallel, or identical. The ratio test tells which without solving.
- ▸The EST usually wants a combination of x and y — add or subtract before you solve.
- ▸With an unknown coefficient, set ratios equal; never try to find x and y.