Digital SAT · Math · Unit 1 · Algebra

Lesson 1.2 · Linear Equations

On the test: the most common question type on the whole test — usually 3–5 per sitting

Big idea. A linear equation is a balance: the left side weighs exactly what the right side weighs. Anything you do to BOTH sides keeps it balanced, so you may keep doing fair moves until x stands alone. The power of x is only ever 1 — that is what ‘linear’ means — so the graph of each side is a straight line.

🎛 Move it · the balance

$3 x + 2 = x + 8$ . Every move is done to BOTH pans, so the scale never tips. Get one purple block alone.

3x + 2 = x + 8

xxx

purple block = one x · gold dot = 1

🧑‍🏫 Rule 1 · Undo with the opposite operationstep 1 of 5

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📋 The question · SAT · March US 2023 · Math 800 Book Q5
What value of $x$ is the solution to the equation $x + 10 = 3$ ?
1. A) $- 13$
2. B) $- 7$
3. C) $7$
4. D) $13$
the idea
$x + 10 = 3$
March US 2023. Something was done to $x$ : 10 was ADDED. To get $x$ back, do the opposite of adding.

🧑‍🏫 Rule 2 · Two layers: undo the outside one firststep 1 of 4

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📋 The question · SAT · October US 2024 · Math 800 Book Q59
$13 x + 50 = 180$
Which equation has the same solution as the given equation?
1. A) $13 x = 50$
2. B) $13 x = 80$
3. C) $13 x = 130$
4. D) $13 x = 180$
the question
$13 x + 50 = 180$
October US 2024. Two things happened to $x$ : it was multiplied by 13, THEN 50 was added.

🧑‍🏫 Rule 3 · Brackets: share the outside number with every inside termstep 1 of 5

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📋 The question · SAT · August US 2023 · Math 800 Book Q10
$6 (x - 9) = 11 (x - 9)$
What value of $x$ is the solution to the given equation?
(type the answer)
the question
$6 (x - 9) = 11 (x - 9)$
August US 2023. A number touching a bracket means MULTIPLY everything inside.

🧑‍🏫 Rule 4 · x on both sides: collect, then solvestep 1 of 4

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📋 The question · SAT · May US 2024 · Math 800 Book Q32
What value of $x$ is the solution to the equation $20 x + 72 = 32 x$ ?
1. A) $- 5$
2. B) $\frac{1}{6}$
3. C) $\frac{6}{13}$
4. D) $6$
the question
$20 x + 72 = 32 x$
May US 2024. $x$ sits on both sides. The goal: all the $x$ ’s on one side, all the plain numbers on the other.

🧑‍🏫 Rule 5 · Fractions: multiply them away firststep 1 of 7

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📋 The question · SAT · August US 2023 · Math 800 Book Q8
$\frac{1}{3} (6 y - 9) = 13$
What value of $y$ satisfies the given equation?
(type the answer)
the question
$\frac{1}{3} (6 y - 9) = 13$
August US 2023. A fraction in front means "divide by 3". Undo it by multiplying by 3.

Why it works · why “both sides”

An equation is a promise that two amounts are equal. Change both by the same amount and the promise still holds. Change one only and you have written a different, false equation.

Trap · the sign that rides with the number

6 (x - 9)

the 6 multiplies

- 9

, giving

- 54

, not

+ 54

. The minus belongs to the 9, so it comes along.

Fast lane · answer the question asked

"What is

12 x

?", "What is

7 p

?", "Which equation has the same solution?" — the answer is often a line you write ON THE WAY. Read the last line of the question before you start.

Desmos move · graph both sides

Type y = 20x + 72 and y = 32x. Tap where they cross: the

x

of that point is the solution (here 6). Works for any equation, even ugly fractions.

Real question 1SAT · August US 2024 · Math 800 Book Q44

\frac{1}{6} (x + 2) - \frac{1}{5} (x + 2) = - 5

What value of $x$ is the solution to the given equation?

Real question 2SAT · March US 2023 · Math 800 Book Q5

What value of $x$ is the solution to the equation $x + 10 = 3$ ?

🎛 Move it · two sides, two lines

The left side $a x + b$ and the right side $c x + d$ are each a line. Slide the numbers. Where the lines MEET is a solution.

2x + 1 = 2x + 4

a2b1c2d4

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

🧑‍🏫 Rule 6 · Simplify both sides, then comparestep 1 of 5

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📋 The question · SAT · March 2024 · Math 800 Book Q21
How many solutions does the equation $9 x + 6 = 5 x + 4 x + 12$ have?
1. A)Exactly one
2. B)Exactly two
3. C)Infinitely many
4. D)Zero
the question
$9 x + 6 = 5 x + 4 x + 12$
March 2024: how many solutions? First make each side as simple as possible.

🧑‍🏫 Rule 7 · “It has no solution — find r”step 1 of 5

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📋 The question · SAT · December 2023 · Math 800 Book Q19
$- 5 x + 15 r x = 135$
In the given equation, $r$ is a constant. The equation has no solution. What is the value of $r$ ?
1. A) $\frac{1}{9}$
2. B) $\frac{1}{3}$
3. C) $3$
4. D) $9$
the question
$- 5 x + 15 r x = 135$
December 2023 (and three more sittings). $r$ is a constant. When does an equation with $x$ only on the LEFT have no solution?

Why it works · why parallel lines mean no solution

Same

x

-number = same steepness. Two lines with the same steepness but starting at different heights stay the same distance apart forever — they never meet, so no

x

makes the sides equal.

Fast lane · match the coefficients

For "no solution": set the

x

-coefficients equal and check that the constants are NOT equal. For "infinitely many": both must be equal. Two tiny equations, no solving.

Trap · 0 = 0 is not “no solution”

x + 1 = x + 1

ends at

1 = 1

— always TRUE, so every

x

works (infinitely many).

x = x + 1

ends at

0 = 1

— always false (none).

Desmos move · see it

Type both sides as lines. Crossing → one. Parallel → none. One line hiding the other → infinitely many.

Real question 3SAT · June 2024 · Math 800 Book Q41

\frac{8 ( c x + 3 )}{5} = 4 x - 3

In the given equation, $c$ is a constant. If the equation has no solution, what is the value of $c$ ?

Real question 4SAT · October US 2024 · Math 800 Book Q60

Which equation has no solution?

🎛 Move it · spot the chunk

November 2024: $8 (x + 7) = 7 (x + 7) + 68$ . What is $x + 7$ ? Tap the chunk.

The question asks for $x + 7$ . Tap where that chunk already sits in the equation.

🧑‍🏫 Rule 8 · A chunk that appears twice is one letterstep 1 of 6

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📋 The question · SAT · March 2024 · Math 800 Book Q18
If $- 9 (15 - 3 x) + 5 = - 10 (15 - 3 x) + 21$ , what is the value of $15 - 3 x$ ?
(type the answer)
the question
$- 9 (15 - 3 x) + 5 = - 10 (15 - 3 x) + 21$
March 2024: what is $15 - 3 x$ ? The same chunk on both sides — rename it.

🎛 Move it · absolute value is distance

$∣ x - h ∣ = k$ says “ $x$ is $k$ steps away from $h$ ”. Slide $h$ and $k$ and watch the two answers.

|x − 6| = 13

-30-20-100102030h = 6

centre h = 6distance k = 13

Two answers: 6 − 13 = -7 and 6 + 13 = 19. Their sum is 12 = 2 × 6 — always twice the centre.

🧑‍🏫 Rule 9 · |stuff| = k splits into two equationsstep 1 of 7

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📋 The question · SAT · August US 2023 · Math 800 Book Q7
$∣3 x - 12∣ = 18$
What is the sum of the solutions to the given equation?
1. A) $- 2$
2. B) $8$
3. C) $10$
4. D) $12$
the question
$∣3 x - 12∣ = 18$
August US 2023: sum of the solutions. $∣ ∣$ means “distance from 0”, so the inside is 18 OR $- 18$ .

Trap · isolate first

∣ x - 9∣ + 81 = 99

: subtract 81 first (

∣ x - 9∣ = 18

). Splitting

x - 9 + 81 = \pm 99

is wrong — the 81 is outside the bars.

Desmos move · absolute value on a graph

Type y = |3x − 12| and y = 18: a V and a flat line, two crossings. Read both

x

’s and add.

Real question 5SAT · December 2024 · Math 800 Book Q86type the answer

If $9 - 4 (4 - 9 x) = 2 - 5 (4 - 9 x)$ , what is the value of $4 - 9 x$ ?

Real question 6SAT · August US 2023 · Math 800 Book Q7

∣3 x - 12∣ = 18

What is the sum of the solutions to the given equation?

🎛 Move it · build the cost

August US 2023: a bus costs $950 for the first 3 hours and $50 for each hour AFTER the first 3. Find the hours that make $1,250.

rent for t = 5 hours

$950
first 3 h

+$50

cost = 950 + 50(5 − 3) = 950 + 50·2 = 1050

The first 3 hours are inside the big block, so only the hours AFTER them get a $50 block: that count is t − 3.

🧑‍🏫 Rule 10 · Translate, one phrase at a timestep 1 of 5

1
find the unknown
$t = hours$
Step 1 is always: what is the letter? Write it down in words.

🧑‍🏫 Rule 11 · Mixtures: amount of the stuff = percent × volumestep 1 of 4

1
📋 The question · SAT · October US 2024 · Math 800 Book Q68
$0.10 x + 0.20 y = 0.19 (x + y)$
The equation gives a volume $x$ , in gallons, of a 10% salt water solution that could be mixed with a volume $y$ , in gallons, of a 20% saltwater solution to produce a 19% saltwater solution. What volume, in gallons, of the 20% saltwater solution could be mixed with 50.0 gallons of 10% saltwater solution to produce a 19% saltwater solution?
(type the answer)
the question
$0.10 x + 0.20 y = 0.19 (x + y)$
October US 2024. 10% salt water ( $x$ gallons) + 20% salt water ( $y$ ) makes 19% salt water.

🧑‍🏫 Rule 12 · Percent increase: multiply by (1 + rate)step 1 of 4

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📋 The question · SAT · August 2024 · Math 800 Book Q53
The result of increasing the quantity $x$ by 600% is 126. What is the value of $x$ ?
1. A)18
2. B)21
3. C)474
4. D)756
the question
$x increased by 600% = 126$
August 2024. “Increase by 600%” means you KEEP the original (100%) and ADD 600%.

🧑‍🏫 Rule 13 · What does a number MEAN in the equation?step 1 of 5

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📋 The question · SAT · October US 2024 · Math 800 Book Q72
To purchase a used car at a total price of 14,000 dollars, a one-time down payment is required, and then fixed monthly payments are made for the remaining amount owed for the car. The equation $14, 000 = 2, 800 + 200 t$ represents this situation, where $t$ is the number of fixed monthly payments that are made. Which of the following is the best interpretation of 200 in this context?
1. A)The amount, in dollars, of the down payment.
2. B)The amount, in dollars, of each fixed monthly payment.
3. C)The total amount, in dollars, paid for the car after $t$ fixed monthly payments.
4. D)The total number of fixed monthly payments.
the question
$14000 = 2800 + 200 t$
October US 2024: car price 14,000, $t$ = number of monthly payments. What is 200?

Why it works · why “after the first 3” gives t − 3

If you rent for 5 hours, the first 3 are inside the flat fee, so only 2 hours are extra — and

5 - 3 = 2

Trap · “each” vs “total”

2.00 x

is a TOTAL (for

x

items). The $2.00 alone is the amount for EACH. Options love to swap these.

Fast lane · plug the options

When the options are equations, invent a case you can do in your head (rent 4 hours → $1,000) and see which equation agrees.

Desmos move · word problem → crossing

Type the left side with

x

for the unknown and the total as a flat line; the crossing is the answer.

Real question 7SAT · October US 2024 · Math 800 Book Q68type the answer

0.10 x + 0.20 y = 0.19 (x + y)

The equation gives a volume $x$ , in gallons, of a 10% salt water solution that could be mixed with a volume $y$ , in gallons, of a 20% saltwater solution to produce a 19% saltwater solution. According to this equation, what volume, in gallons, of the 20% saltwater solution could be mixed with 50.0 gallons of 10% saltwater solution to produce a 19% saltwater solution?

Real question 8SAT · October US 2024 · Math 800 Book Q72

To purchase a used car at a total price of 14,000 dollars, a one-time down payment is required, and then fixed monthly payments are made for the remaining amount owed for the car. The equation $14, 000 = 2, 800 + 200 t$ represents this situation, where $t$ is the number of fixed monthly payments that are made. Which of the following is the best interpretation of 200 in this context?

🧑‍🏫 Rule · A proportion: cross-multiplystep 1 of 3

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📋 The question · Gen Z Study
If $\frac{x + 2}{6} = \frac{x - 1}{4}$ , what is the value of $x$ ?
(type the answer)
cross-multiply
$4 (x + 2) = 6 (x - 1)$
When one fraction equals another, top-left × bottom-right = top-right × bottom-left. (It is multiplying both sides by both denominators.)

🧑‍🏫 Rule · Decimals: multiply every term by 10, 100, …step 1 of 2

1
📋 The question · Gen Z Study
What value of $x$ satisfies $0.3 x + 1.2 = 0.15 x + 2.1$ ?
(type the answer)
clear decimals
$30 x + 120 = 15 x + 210$
The most decimal places is 2, so multiply EVERY term by 100.

Take with you

▸Every move is done to BOTH sides; undo operations in reverse order (last on, first off).
▸Brackets: the outside number multiplies EVERY inside term, sign included. Fractions: multiply every term by the common denominator first.
▸Same $x$ -part and different numbers → no solution; same everything → infinitely many; different $x$ -parts → exactly one.
▸“No solution, find r”: make the $x$ -coefficients equal (the constant parts must differ).
▸A chunk that repeats is one letter; the same chunk over two different numbers must be 0.
▸ $∣ x - h ∣ = k$ : two answers $h \pm k$ , summing to $2 h$ ; $k = 0$ gives one, $k < 0$ none. Isolate the bars first.
▸Words: fixed part + rate × (count). A number next to the letter is a per-item rate; a lone number is one-time.

Keep reading — this lesson marks itself done automatically.