Digital SAT · Math · Unit 1 · Algebra

Lesson 1.3 · Systems of Linear Equations

On the test: 2–4 questions a test; many are 30-second questions if you spot the shortcut

Big idea. A system is two (or more) equations that must be true AT THE SAME TIME. Each linear equation is a line, so the solution is the point that sits on both lines — where they cross. Every method (substitution, elimination, graphing) is just a way of finding that crossing point.

🎛 Move it · two lines, one crossing

Each side here is one line of a system $y = a x + b$ and $y = c x + d$ . The gold point is the solution: the one $(x, y)$ that makes BOTH equations true.

2x + 1 = 2x + 4

a2b1c2d4

Same x-number, different constants → parallel lines that never meet → NO solution.

🧑‍🏫 Rule 1 · A solution satisfies BOTH equationsstep 1 of 3

1
the meaning
$(x, y) works in eq. 1 AND eq. 2$
One equation alone has infinitely many solutions (every point on its line). The second equation narrows it down to the points on BOTH lines.

🧑‍🏫 Rule 2 · Substitution: use what one equation already tells youstep 1 of 4

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📋 The question · SAT · March 2024 · Math 800 Book Q12
$3 x - 4 y = 5$ and $x = 7$ . What is the solution $(x, y)$ to the given system of equations?
1. A) $(7, 4)$
2. B) $(7, 12)$
3. C) $(7, - 12)$
4. D) $(7, - 4)$
the question
$3 x - 4 y = 5, x = 7$
March 2024. The second equation hands you $x$ directly.

🧑‍🏫 Rule 3 · Both start with “y =”? Set them equalstep 1 of 5

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📋 The question · SAT · October US 2024 · Math 800 Book Q40
$y = 5 x + 7$ and $y = - 4 x - 11$ . The solution is $(x, y)$ . What is the value of $y$ ?
1. A) $- 18$
2. B) $- 4$
3. C) $- 3$
4. D) $- 2$
the question
$y = 5 x + 7, y = - 4 x - 11$
October US 2024. Both right sides equal the same $y$ , so they equal each other.

Why it works · why substitution works

x = 7

is true, then

x

and 7 are the same number, so swapping one for the other cannot change whether an equation is true.

Desmos move · type both lines

Type each equation exactly as given (Desmos accepts 3x − 4y = 5). Tap the crossing point — Desmos shows its coordinates. That IS the solution.

Real question 1SAT · October 2024 · Math 800 Book Q41

The solution to the system $y = 6 x + 16$ and $- 7 x - y = 36$ is $(x, y)$ . What is the solution?

🎛 Move it · the elimination machine

March US 2023: $5 x + 4 y = 44$ and $3 x - 7 y = 17$ . What is $8 x - 3 y$ ? Look at what ONE copy of each, added, already gives. Then try making $x$ or $y$ vanish.

1×

(5 x + 4 y = 44)

1×

(3 x - 7 y = 17)

8 x - 3 y = 61

✓ That is exactly the expression the question asked for — no need to find x or y at all.

🧑‍🏫 Rule 4 · Adding two true equations gives a true equationstep 1 of 4

1
why it is allowed
$A = B and C = D \Rightarrow A + C = B + D$
Equal amounts added to equal amounts stay equal — two balanced scales poured onto one scale still balance.

🧑‍🏫 Rule 5 · Make the numbers opposites, then addstep 1 of 5

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📋 The question · SAT · December 2024 · Math 800 Book Q56
If $x + 4 y = 41$ and $7 x - 20 y = - 97$ , what is the value of $y$ ?
(type the answer)
the question
$x + 4 y = 41, 7 x - 20 y = - 97$
December 2024: find $y$ . No letter cancels yet.

🧑‍🏫 Rule 6 · Chunks in a system: treat each chunk as a letterstep 1 of 3

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📋 The question · SAT · August US 2024 · Math 800 Book Q37
$(x - 2) - 8 (y + 3) = 123$ and $(x - 2) + 8 (y + 3) = 432$ . The solution is $(x, y)$ . What is the value of $10 (x - 2)$ ?
(type the answer)
the question
$(x - 2) - 8 (y + 3) = 123, (x - 2) + 8 (y + 3) = 432$
August US 2024: what is $10 (x - 2)$ ? The chunks $(x - 2)$ and $8 (y + 3)$ appear in both.

Fast lane · look at what is asked FIRST

8 x - 3 y

14 y

9 x + 8 y

? Add or subtract the equations as they stand and see if it appears. On the real test it usually does.

Trap · multiply every term

Multiplying only the left side breaks the balance.

5 (x + 4 y) = 5 \cdot 41

, not 41.

Real question 2SAT · March US 2023 · Math 800 Book Q1type the answer

$5 x + 4 y = 44$ and $3 x - 7 y = 17$ . The solution to the given system of equations is $(x, y)$ . What is the value of $8 x - 3 y$ ?

Real question 3SAT · August 2024 · Math 800 Book Q33type the answer

$\frac{x - 18}{6} + \frac{y + 25}{3} = - 5$ and $\frac{x - 18}{3} - \frac{y + 25}{6} = 5$ . The solution to the given system of equations is $(x, y)$ . What is the value of $x - y$ ?

🧑‍🏫 Rule 7 · Compare the two lines’ ratiosstep 1 of 5

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standard form
$a_{1} x + b_{1} y = c_{1}, a_{2} x + b_{2} y = c_{2}$
Write both equations in the same shape.

🧑‍🏫 Rule 8 · Find the constantstep 1 of 4

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📋 The question · SAT · August US 2023 · Math 800 Book Q6
$2 x + y = 8$ and $a x + 3 y = 24$ . In the given system of equations, $a$ is a constant. The system has infinitely many solutions. What is the value of $a$ ?
(type the answer)
the question
$2 x + y = 8, a x + 3 y = 24$
August US 2023: infinitely many solutions. What is $a$ ?

🧑‍🏫 Rule 9 · “For each real number r, which point lies on both?”step 1 of 5

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📋 The question · SAT · March US 2023 · Math 800 Book Q4
$3 x + 5 y = 8$ and $9 x + 15 y = 24$ . For each real number $r$ , which of the following points lies on the graph of each equation in the $x y$ -plane for the given system?
1. A) $(r, - \frac{5 r}{3} + \frac{8}{3})$
2. B) $(r, \frac{3 r}{5} + \frac{8}{5})$
3. C) $(- \frac{5 r}{3} + \frac{8}{3}, r)$
4. D) $(\frac{r}{3} + 8, - \frac{r}{3} + 24)$
the question
$3 x + 5 y = 8, 9 x + 15 y = 24$
June 2023. The second is 3 × the first: the SAME line, infinitely many points. The question asks for a formula for every one of them.

🧑‍🏫 Rule 10 · Three equations at oncestep 1 of 4

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📋 The question · SAT · March 2024 · Math 800 Book Q17
$x + 4 y = - 16$ , $5 x + 20 y = 0$ , $3 x + 12 y = 34$ . The given equations are graphed in the $x y$ -plane. How many solutions $(x, y)$ does the system of three equations have?
1. A)Zero
2. B)Exactly one
3. C)Exactly two
4. D)Infinitely many
the question
$x + 4 y = - 16, 5 x + 20 y = 0, 3 x + 12 y = 34$
March 2024: how many solutions does the system of THREE equations have?

Desmos move · parallel or same?

Type both equations. Two visible lines that never meet → none. You only see one line (they overlap) → infinitely many.

Real question 4SAT · August 2024 · Math 800 Book Q35

$figure from the test$

Figure from the test paper — our labels appear after you answer.

The graph of a system of two linear equations is shown. If a third equation, $x + 2 y = - 8$ , is added to this system, how many solutions will the resulting system of three equations have?

🧑‍🏫 Rule 11 · Two unknowns, two sentences, two equationsstep 1 of 5

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📋 The question · SAT · March 2024 · Math 800 Book Q21
Two customers bought the same kind of cheese and milk at a store. The first customer paid 12.45 dollars for 1 pound of cheese and 2 gallons of milk. The second customer paid 19.70 dollars for 3 pounds of cheese and 1 gallon of milk. What is the cost, in dollars, of 1 gallon of milk?
1. A)3.53
2. B)4.15
3. C)4.93
4. D)5.39
the question
$cheese c, milk m$
March 2024: customer 1 paid $12.45 for 1 lb cheese + 2 gal milk; customer 2 paid $19.70 for 3 lb cheese + 1 gal milk. Name the two unknown PRICES.

🧑‍🏫 Rule 12 · Mixtures: one equation for the total, one for the ingredientstep 1 of 4

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📋 The question · SAT · March 2024 · Math 800 Book Q61
At a constant temperature, a scuba diver combines two different concentrations of nitrox mix to fill a tank to a total pressure of 130 atm, of which 34% is contributed by oxygen. Mix A contributes $x$ atm, of which 22% is oxygen. Mix B contributes $y$ atm, of which 48% is oxygen. Which system represents this situation?
1. A) $x + y = 130$ and $0.48 x + 0.22 y = 0.34 (130)$
2. B) $x + y = 130$ and $0.22 x + 0.48 y = 0.34 (130)$
3. C) $x + y = 0.34 (130)$ and $0.48 x + 0.22 y = 130$
4. D) $x + y = 0.34 (130)$ and $0.22 x + 0.48 y = 130$
the question
$tank 130 atm, 34% oxygen$
March 2024: mix A gives $x$ atm (22% oxygen), mix B gives $y$ atm (48% oxygen).

🧑‍🏫 Rule 13 · A line and a curve: substitute, then countstep 1 of 4

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📋 The question · SAT · March 2024 · Math 800 Book Q19
$y = 6 x^{2} - 36 x + 56$ and $y + 3 = 0$ . How many solutions are there to the given system of equations?
1. A)There is exactly 1 solution.
2. B)There are exactly 2 solutions.
3. C)There are exactly 3 solutions.
4. D)There are no solutions.
the question
$y = 6 x^{2} - 36 x + 56, y + 3 = 0$
March 2024: how many solutions? The second says $y = - 3$ , a flat line.

Fast lane · the question picks the letter

If it asks for the price of milk, eliminate cheese. Fewer steps, fewer slips.

Desmos move · line meets curve

Type both. Count the crossings — that is the number of solutions; tap each to read it.

Real question 5SAT · December 2024 · Math 800 Book Q54

$figure from the test$

Figure from the test paper — our labels appear after you answer.

The graph of a system of two linear equations is shown. The solution to the system is $(x, y)$ . What is the value of $y$ ?

Real question 6SAT · March US 2026 (Form A) · Module 1 · Q4

$figure from the test$

Figure from the test paper — our labels appear after you answer.

The graph of a system of a linear equation and a quadratic equation is shown. What is the solution $(x, y)$ to this system?

🧑‍🏫 Rule · Which method is faster?step 1 of 4

1
substitution
$x = \dots or y = \dots already$
If one equation is already solved for a letter (or has a letter with coefficient 1), substitute.

Take with you

▸A solution of a system is the point on EVERY graph — where the lines cross.
▸One letter given? Substitute it. Both “ $y =$ ”? Set them equal.
▸Elimination: make one letter’s numbers opposites (multiply every term), then add. Always check first whether adding/subtracting as-is gives what is asked.
▸Ratios $a_{1} : a_{2}$ vs $b_{1} : b_{2}$ vs $c_{1} : c_{2}$ : different → one; same left, different right → none; all same → infinitely many.
▸“For each $r$ ” → the two equations are the same line; plug $r = 0$ into the options and test.
▸Words: one equation per fact (total + ingredient, or one per purchase). Line meets curve: substitute and count, or count crossings on the graph.

Keep reading — this lesson marks itself done automatically.