PSAT 10 Linear Equations, Functions & Systems Quick Review

What You Need to Know

You’ll see linear equations, functions, and systems all over the PSAT 10 because they test whether you can model relationships, read graphs, and solve for unknowns efficiently.

Big ideas (the stuff PSAT loves)

• Linear equation: variables are only to the first power. Graph is a line.
• Function: a rule that assigns each input exactly one output (think “machine”).
• System: two (or more) equations at the same time; solutions are the values that satisfy all of them.

What you’re expected to do quickly

• Solve linear equations in one variable and interpret solutions.
• Rearrange formulas (solve for a variable).
• Understand and use slope, intercepts, and different line forms.
• Interpret function notation and key features from tables/graphs.
• Solve systems by substitution or elimination, including “no solution” and “infinitely many solutions.”

Critical reminder: The PSAT frequently tests meaning, not just computation: what does $m$ mean in context, what does an intersection point represent, what does $f(2)$ mean, etc.

Step-by-Step Breakdown

A) Solving linear equations (one variable)

1. Distribute to remove parentheses.
2. Combine like terms on each side.
3. Move variable terms to one side (add/subtract).
4. Isolate the variable (multiply/divide).
5. Check quickly if the result seems reasonable (especially word problems).

Mini example
Solve:
$3(2x-5)=4x+1$
Distribute:
$6x-15=4x+1$
Move $4x$ left:
$2x-15=1$
Add $15$:
$2x=16$
Divide by $2$:
$x=8$

B) Rearranging formulas (solve for a variable)

1. Identify what you’re solving for.
2. Undo operations step-by-step (reverse PEMDAS).
3. Keep the equation balanced (do the same thing to both sides).
4. Factor the variable out if it appears in multiple terms.

Mini example
Solve for $y$:
$A=2x+3y$
Subtract $2x$:
$A-2x=3y$
Divide by $3$:
$y=\frac{A-2x}{3}$

C) Lines: build an equation from information

If you know slope and a point

1. Compute slope if needed:
   $m=\frac{y_2-y_1}{x_2-x_1}$
2. Use point-slope:
   $y-y_1=m(x-x_1)$
3. Convert to slope-intercept if helpful:
   $y=mx+b$

Mini example (two points)
Points: $(-1,2)$ and $(3,10)$
$m=\frac{10-2}{3-(-1)}=\frac{8}{4}=2$
Use $(-1,2)$:
$y-2=2(x+1)$
Simplify:
$y=2x+4$

D) Function questions (notation + reading)

1. Translate: $f(2)$ means “plug $2$ in for $x$.”
2. Read from a graph: $f(2)$ is the **y-value** where $x=2$.
3. A function is linear if the rate of change is constant (same slope between points).

Mini example
If:
$f(x)=3x-7$
Then:
$f(2)=3(2)-7=-1$

E) Systems of linear equations

Method 1: Substitution (best when one equation is already solved for a variable)

1. Solve one equation for one variable.
2. Substitute into the other equation.
3. Solve the resulting one-variable equation.
4. Plug back to find the other variable.

Mini example
$y=x+2$
$2x+y=11$
Substitute:
$2x+(x+2)=11$
$3x=9$
$x=3$
Then:
$y=3+2=5$
Solution: $(3,5)$

Method 2: Elimination (best when coefficients line up or can be made to line up)

1. Align equations in $Ax+By=C$ style.
2. Multiply one or both equations so one variable cancels when added/subtracted.
3. Add/subtract to eliminate.
4. Solve, then back-substitute.

Mini example
$3x+2y=18$
$3x-2y=6$
Add:
$6x=24$
$x=4$
Plug in:
$3(4)+2y=18$
$12+2y=18$
$y=3$
Solution: $(4,3)$

Special system outcomes (PSAT trap zone)

• No solution: parallel lines (same slope, different intercept). You’ll get something like:
  $0=5$
• Infinitely many solutions: same line (equations are multiples). You’ll get something like:
  $0=0$

Key Formulas, Rules & Facts

Linear equations and lines

Intercepts and meaning

Functions

Systems

Examples & Applications

Example 1: Interpret slope and intercept in context

A gym charges a membership fee plus a monthly cost. Total cost after $m$ months is:
$C=15m+40$

• Slope: $15$ means $\$15$ per month.
• Intercept: $40$ means $\$40$ initial fee when $m=0$.
• Cost after $6$ months:
  $C=15(6)+40=130$

Example 2: Build a line from a word description

“Temperature drops $3$ degrees per hour. At $t=0$, it’s $72$ degrees.”

• Rate (slope) $m=-3$
• Intercept $b=72$
  Equation:
  $T=-3t+72$
  When will it hit $60$ degrees?
  $60=-3t+72$
  $-12=-3t$
  $t=4$

Example 3: Function from a table (constant rate check)

Table points: $(1,5),(3,9),(5,13)$

• Slope from first two:
  $m=\frac{9-5}{3-1}=\frac{4}{2}=2$
• Slope from last two:
  $m=\frac{13-9}{5-3}=\frac{4}{2}=2$
  Constant slope $\Rightarrow$ linear.
  Use $(1,5)$ in $y=mx+b$:
  $5=2(1)+b$
  $b=3$
  So:
  $f(x)=2x+3$

Example 4: System modeling (intersection meaning)

Tickets: Adult $\$12$, Student $\$8$. Total $50$ tickets sold for $\$520$.
Let $a$ adults, $s$ students.
$a+s=50$
$12a+8s=520$
Substitute $s=50-a$:
$12a+8(50-a)=520$
$12a+400-8a=520$
$4a=120$
$a=30$
Then:
$s=20$
Interpretation: intersection gives the only combination that meets both constraints.

Common Mistakes & Traps

1. Sign errors when distributing: You forget to distribute a negative.

   • Wrong: $-(x-4)=-x-4$
   • Correct: $-(x-4)=-x+4$
   • Fix: treat $-1$ like a multiplier.

2. Mixing up slope and intercept: You read $y=mx+b$ but call $b$ the slope.

   • Fix: memorize: $m$ “moves” the line’s steepness; $b$ is where it “begins” on the $y$-axis.

3. Slope from two points with reversed subtraction: You do $\frac{y_1-y_2}{x_2-x_1}$ accidentally.

   • Why it matters: you can flip both numerator and denominator and still be correct, but flipping only one changes the sign.
   • Fix: use a consistent order: $\frac{y_2-y_1}{x_2-x_1}$.

4. Forgetting what $f(x)$ means: You treat $f(x)$ like $f\times x$.

   • Fix: read $f(x)$ as “output of function $f$ at input $x$.”

5. Not checking for special system cases: You stop after elimination gives a weird statement.

   • If you get $0=0$: infinitely many solutions.
   • If you get $0=5$: no solution.

6. Dropping a variable during elimination: You multiply one equation but forget to multiply every term.

   • Fix: multiply the entire equation (both sides).

7. Confusing solution of a system with separate solutions: You solve each equation alone and list two answers.

   • Fix: a system solution is an ordered pair $(x,y)$ satisfying both.

8. Graph interpretation mix-ups: You give an $x$-value when asked for $f(x)$.

   • Fix: $f(x)$ is always a $y$-value (output).

Memory Aids & Quick Tricks

Quick Review Checklist

• You can solve linear equations by distribute → combine → isolate.
• You can rearrange formulas to solve for any variable without breaking equality.
• You can compute slope with $m=\frac{y_2-y_1}{x_2-x_1}$ and interpret it as a rate.
• You can write line equations in $y=mx+b$ or $y-y_1=m(x-x_1)$.
• You can find intercepts by setting $x=0$ (for $y$-intercept) or $y=0$ (for $x$-intercept).
• You understand $f(a)$ as an output and can read it from a table or graph.
• You can solve systems by substitution and elimination, and identify:
  • one solution (intersection)
  • no solution (parallel)
  • infinitely many solutions (same line)

You’ve got this: focus on clean algebra, and always attach meaning to what you’re solving for.