SAT Algebra: linear equations, functions and systems

In this lesson you will learn

Decide whether a linear equation has one, no or infinitely many solutions
Interpret slope and y-intercept in context
Solve systems of two linear equations quickly
Set up and solve linear inequalities in word problems

The most common domain in SAT Math is Algebra: about 13–15 of the 44 questions. These questions are not hard, but they are full of traps: one sign error or missing that the question asks for a different quantity costs you points. In this lesson you will learn to handle linear equations, lines and systems at SAT speed.

1
What is asked?
Read the end of the question and write the target on scratch paper: x, x + y or the number of solutions.
2
Pick a method
Algebra, plugging in the choices (backsolving) or Desmos — whichever is fastest.
3
Solve
Write each step on its own line: most sign errors happen “in your head”.
4
Check
Put the answer back into the original equation and make sure you give the quantity asked for.

Linear equations in one variable

ax + b = cx + d

where:

a ≠ cone solution: x = (d − b)/(a − c)
a = c, b ≠ dno solution
a = c, b = dinfinitely many solutions

After expanding brackets and collecting like terms, compare the coefficients of x and the constants.

Example 1: an equation with no solution

For what value of k does the equation 3(x − 2) + kx = 5x + 4 have no solution?

Show solution

Expand the left side: 3x − 6 + kx = (3 + k)x − 6.
Equation: (3 + k)x − 6 = 5x + 4.
For no solution, the x-coefficients must be equal and the constants different: 3 + k = 5 and −6 ≠ 4.
So k = 2. Check: 5x − 6 = 5x + 4 ⇒ −6 = 4 — impossible.

Lines: slope and starting value

m = (y₂ − y₁) / (x₂ − x₁) y = mx + bm = (y₂ − y₁) / (x₂ − x₁) y = mx + b

where:

mslope — the change in y when x increases by 1: a rate or a price per unit
by-intercept — the value of y when x = 0: the starting value
(x₁, y₁), (x₂, y₂)two points on the line

Parallel lines have equal slopes; for perpendicular lines m₁ · m₂ = −1.

Example 2: interpreting a model

Aysel's savings are modeled by S = 25w + 120, where S is the amount in manat and w is the number of weeks. What do 25 and 120 represent? After how many weeks will Aysel have 370 manat?

Show solution

25 is the slope: Aysel adds 25 manat every week.
120 is the starting value: at w = 0 she already has 120 manat.
25w + 120 = 370 ⇒ 25w = 250 ⇒ w = 10 weeks.

Interactive

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At first the lines y = x + 1 and y = −0.5x + 4 meet at (2, 3) — the solution of the system. Set m to −0.5: the lines become parallel and the system has no solution. Then set b to 4: the lines coincide — infinitely many solutions.

Systems of two equations

a₁x + b₁y = c₁, a₂x + b₂y = c₂

where:

a₁/a₂ ≠ b₁/b₂a₁/a₂ ≠ b₁/b₂one solution — the lines intersect
a₁/a₂ = b₁/b₂ ≠ c₁/c₂a₁/a₂ = b₁/b₂ ≠ c₁/c₂no solution — the lines are parallel
a₁/a₂ = b₁/b₂ = c₁/c₂a₁/a₂ = b₁/b₂ = c₁/c₂infinitely many solutions — the same line

Example 3: elimination

The solution to the system 2x + 3y = 12 and 4x − 3y = 6 is (x, y). What is the value of x + y?

Show solution

The y-coefficients are opposites (+3y and −3y), so add the equations: 6x = 18 ⇒ x = 3.
Substitute into the first equation: 2 · 3 + 3y = 12 ⇒ 3y = 6 ⇒ y = 2.
x + y = 3 + 2 = 5.
Careful: the question asks for x + y, not x — the answer 3 is a trap.

Linear inequalities

Example 4: “at most”

Murad has 50 manat. He buys a notebook for 7 manat and pens that cost 4 manat each. What is the greatest number of pens he can buy?

Show solution

Let p be the number of pens: 7 + 4p ≤ 50.
4p ≤ 43 ⇒ p ≤ 10.75.
The number of pens is a whole number no greater than 10.75, so 10 pens.
Round down, not up: 11 pens would cost 51 manat.

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Key points

ax + b = cx + d: if a = c and b ≠ d there is no solution; if a = c and b = d there are infinitely many.
y = mx + b: m is the rate of change, b is the starting value.
In a system, the ratios of the coefficients reveal the number of solutions.
If the question asks for an expression (x + y, 2x + 3y), add or subtract the equations directly.
Flip the inequality sign when multiplying or dividing by a negative number.

Check yourself

10 questions. Every correct answer earns XP.

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If 4x − 7 = 2x + 9, what is the value of x?