PSAT 8/9 Math — Algebra: Linear Relationships and Inequalities

Linear equations in one variable

A linear equation in one variable is an equation where the variable (usually $x$) only appears to the first power, and the solution is a single number that makes the equation true. “Linear” means the graph would be a straight line if you graphed it, but in one-variable equations you’re usually thinking of a balance: whatever you do to one side, you must do to the other to keep the equation true.

This topic matters because it is the core skill behind almost everything else in algebra: solving for an unknown, rearranging formulas, and checking whether a value satisfies a condition. On the PSAT 8/9, you’ll often solve one-step, two-step, and multi-step linear equations, including those with parentheses and fractions.

What “solving” really means

To solve an equation means to find the value of the variable that makes the left side equal to the right side. A useful mindset is:

• The equals sign means “these two expressions have the same value.”
• Your job is to rewrite the equation into an equivalent form where the value of the variable becomes obvious.

The main tools are the properties of equality:

• Add/subtract the same number on both sides.
• Multiply/divide by the same nonzero number on both sides.

How to solve step by step (the mechanism)

Most linear equations can be solved by “undoing” operations in reverse order.

1. Simplify each side: distribute, combine like terms.
2. Collect variable terms on one side (often by adding or subtracting a term).
3. Collect constants on the other side.
4. Isolate the variable by dividing or multiplying.
5. Check by substitution (especially if negatives or fractions are involved).

A key idea: when you move a term across the equals sign, you are really adding or subtracting that term from both sides.

Worked example 1: variables on both sides

Solve:
$3x - 7 = x + 9$

Step 1: get variable terms on one side by subtracting $x$ from both sides:
$3x - x - 7 = 9$

Step 2: combine like terms:
$2x - 7 = 9$

Step 3: add $7$ to both sides:
$2x = 16$

Step 4: divide by $2$:
$x = 8$

Check: left side $3(8) - 7 = 24 - 7 = 17$, right side $8 + 9 = 17$, so it works.

Worked example 2: parentheses (distribution)

Solve:
$4(2x - 3) = 3x + 6$

Distribute:
$8x - 12 = 3x + 6$

Subtract $3x$ from both sides:
$5x - 12 = 6$

Add $12$:
$5x = 18$

Divide:
$x = \frac{18}{5}$

Worked example 3: fractions (clear denominators carefully)

Solve:
$\frac{x}{3} + 2 = 7$

Subtract $2$:
$\frac{x}{3} = 5$

Multiply both sides by $3$:
$x = 15$

A common mistake is multiplying only one term by $3$. You must multiply the entire side.

Special cases: no solution vs infinitely many solutions

Sometimes simplifying shows something surprising.

• No solution happens when you get a false statement.
• Infinitely many solutions happens when you get a true statement.

Example (no solution):
$2x + 1 = 2x + 5$
Subtract $2x$:
$1 = 5$
That can’t be true, so there is no solution.

Example (infinitely many):
$3(x + 2) = 3x + 6$
Distribute:
$3x + 6 = 3x + 6$
This is always true, so every $x$ works.

Exam Focus

• Typical question patterns:
  • Solve multi-step equations with variables on both sides (often with distribution).
  • Solve equations that represent a real situation (prices, distances, ages).
  • Decide whether an equation has one solution, no solution, or infinitely many solutions.
• Common mistakes:
  • Distributing incorrectly, especially with negatives (e.g., treating $-(x+3)$ as $-x+3$ instead of $-x-3$).
  • “Moving” terms without doing the same operation to both sides (forgetting the balance idea).
  • Clearing denominators incorrectly by multiplying only one term instead of the entire side.

Linear equations in two variables

A linear equation in two variables (often $x$ and $y$) describes a relationship between the variables where each variable is to the first power. The big idea is that instead of one solution, you usually have **many ordered pairs** \,(x, y)\,$ that make the equation true. Those solutions form a straight line when graphed. This matters because two-variable linear equations are how you model relationships between quantities: total cost as a function of items bought, distance as a function of time, or scoring rules with two types of points. ### Standard forms you’ll see A very common form is **standard form**: Ax + By = C where A$,$B$, and$C are constants (usually integers on tests). Another common form is **slope-intercept form**: y = mx + b where m$is the slope and$b$is the$y-intercept. You don’t need to memorize names as much as you need to recognize what each form makes easy: | Form | Looks like | What it makes easy | |---|---|---| | Standard | Ax + By = C | Checking integer solutions; combining equations in systems | | Slope-intercept | y = mx + b | Graphing quickly; reading slope and intercept | ### Solutions as ordered pairs To check whether an ordered pair is a solution, substitute x$and$y into the equation. Example: Is \,(2, 5)\,$ a solution of
$y = 2x + 1$
Substitute $x = 2$:
$y = 2(2) + 1 = 5$
It matches $y = 5$, so yes.

A common misconception is thinking the “solution” is a single number as in one-variable equations. Here, the “answer” is typically a set of ordered pairs, a line on a graph, or sometimes a specific pair if another condition is added.

Finding points on the line (table approach)

A practical method is to pick convenient $x$ values and compute $y$.

Example: For
$2x + y = 8$
Solve for $y$:
$y = 8 - 2x$
Choose values:

• If $x = 0$, then $y = 8$.
• If $x = 4$, then $y = 0$.

Those two points are often enough to graph the line.

Graphing by intercepts (fast and reliable)

For
$Ax + By = C$

• The $x$-intercept happens when $y = 0$.
• The $y$-intercept happens when $x = 0$.

Example:
$3x + 2y = 12$
$x$-intercept: set $y = 0$:
$3x = 12$
$x = 4$
$y$-intercept: set $x = 0$:
$2y = 12$
$y = 6$
So the line goes through \,(4, 0)\,$ and \,(0, 6)\,$.

Exam Focus

• Typical question patterns:
  • Identify whether a point satisfies a linear equation.
  • Convert between forms (especially rewriting $Ax + By = C$ as $y = mx + b$).
  • Graph a line from an equation or write an equation from a graph.
• Common mistakes:
  • Mixing up intercepts (setting the wrong variable to $0$).
  • Sign errors when solving for $y$ (e.g., mishandling $-2x$).
  • Treating a two-variable equation like it has only one numeric solution.

Linear functions

A linear function is a function whose rule creates a straight-line graph. Functions are special because they come with an input-output structure: each input $x$ has exactly one output $y$.

This matters because functions are a major language of algebra. On the PSAT 8/9, you’ll be expected to interpret linear functions from equations, tables, or graphs, and connect key features like slope and intercepts to meaning in context.

Function notation and what it tells you

Instead of writing $y = mx + b$, you might see:
$f(x) = mx + b$
This means: “The output of the function $f$ when the input is $x$ equals $mx + b$.” It’s the same relationship as $y = mx + b$, but emphasizes the process.

Example: If
$f(x) = 3x - 2$
then
$f(4) = 3(4) - 2 = 10$
A common mistake is to read $f(x)$ as $f \times x$. It does not mean multiplication.

Slope as rate of change

For a linear function
$y = mx + b$

• $m$ is the **slope**, the amount $y$ changes when $x$ increases by $1$.
• $b$ is the $y$-intercept, the value of $y$ when $x = 0$.

If you have two points \,(x_1, y_1)\,$ and \,(x_2, y_2)\,$ on a line, then slope is:
$m = \frac{y_2 - y_1}{x_2 - x_1}$
This formula matters because many questions give you two points or a graph and ask for the equation.

Interpreting slope and intercept in context

Suppose the cost of renting a bike is a starting fee plus a per-hour fee. A model might be:
$C = 5 + 2h$
Here:

• $5$ is the starting cost (when $h = 0$).
• $2$ is the cost per hour (each extra hour increases cost by $2$).

That interpretation skill is tested often: you’re not just solving; you’re explaining what numbers mean.

Writing a linear function from points

If you know a point and the slope, you can build the equation.

Example: A line has slope $m = -3$ and passes through \,(2, 7)\,$. Start with slope-intercept form: y = -3x + b Substitute the point to find b: 7 = -3(2) + b 7 = -6 + b b = 13 So: y = -3x + 13 If you know two points, find the slope first, then proceed. Example: Through \,(1, 2)\,$ and %%LATEXBLOCK117%%m = \frac{10 - 2}{5 - 1} = \frac{8}{4} = 2%%LATEXBLOCK118%%\,(1, 2)\,$:
$y = 2x + b$
$2 = 2(1) + b$
$b = 0$
So:
$y = 2x$

Recognizing linearity from a table

A table represents a linear function if the change in output is constant when the change in input is constant.

Example: If $x$ increases by $1$ each step and $y$ increases by $4$ each step, the slope is $4$.

A common trap: constant ratios do not indicate a linear function; constant differences do.

Exam Focus

• Typical question patterns:
  • Interpret $m$ and $b$ in a real-world model.
  • Find slope from two points, a table, or a graph.
  • Write the equation of a line given points or slope and intercept.
• Common mistakes:
  • Reversing differences in the slope formula (mixing up $\frac{\Delta x}{\Delta y}$ instead of $\frac{\Delta y}{\Delta x}$).
  • Confusing slope with intercept (especially when reading graphs).
  • Misreading function notation, like thinking $f(3)$ means $f \times 3$.

Systems of two linear equations in two variables

A system of two linear equations in two variables is two linear relationships that are both true at the same time. The solution is the ordered pair \,(x, y)\,$ that satisfies both equations. Graphically, it’s the point where the two lines intersect. This matters because real situations often involve multiple constraints at once: a budget and a quantity rule, two pricing plans, or mixtures with two ingredients. On the PSAT 8/9, you may solve systems algebraically or interpret solutions from graphs and contexts. ### What solutions can look like (three possibilities) Two lines in a plane can: 1. Intersect once: **one solution**. 2. Be parallel: **no solution**. 3. Be the same line: **infinitely many solutions**. This connects directly to the “special cases” you saw in one-variable equations: sometimes the constraints contradict each other (no solution) or are redundant (infinitely many solutions). ### Solving by substitution **Substitution** works well when one equation is already solved for a variable. Example: y = x + 3 2x + y = 12 Substitute y = x + 3 into the second equation: 2x + (x + 3) = 12 Combine like terms: 3x + 3 = 12 Subtract 3: 3x = 9 x = 3 Now plug back into y = x + 3: y = 6 Solution: \,(3, 6) Common error: forgetting to substitute the entire expression, especially when parentheses are needed. ### Solving by elimination (addition/subtraction) **Elimination** works well when variables line up. Example: 3x + 2y = 16 3x - 2y = 8 Add the equations to eliminate y: 6x = 24 x = 4 Substitute into one equation: 3(4) + 2y = 16 12 + 2y = 16 2y = 4 y = 2 Solution: \,(4, 2) If coefficients don’t match, you can multiply one or both equations by a constant first. That’s not “changing” the equation; multiplying both sides by the same nonzero number keeps it equivalent. ### Interpreting a system in context Example: Tickets to a school event cost \$5%%LATEXBLOCK156%%\$8$for adults. If$x$student tickets and$y$adult tickets were sold, and$x + y = 50$tickets total, with total revenue$5x + 8y = 340, then the system is: x + y = 50 5x + 8y = 340 Solving tells you how many of each type were sold. The meaning of the solution is often as important as the algebra. ### Checking your solution Always verify by substitution into both equations. This catches sign mistakes and arithmetic slips. #### Exam Focus - **Typical question patterns**: - Solve a system by substitution or elimination and interpret the ordered pair. - Match a system to a word problem (define variables correctly). - Determine the number of solutions from a graph or from equation structure (parallel vs same line). - **Common mistakes**: - Eliminating incorrectly (adding when you should subtract, or mishandling negatives). - After finding x$, forgetting to find$y (or plugging back into the wrong equation form). - Not interpreting the solution in context (e.g., giving y as “dollars” when it represents “tickets”). ## Linear inequalities in one or two variables A **linear inequality** compares expressions using symbols like <$,$>$,$\le$, and$\ge. Instead of one solution, the solution is a **set of values** that make the inequality true. Inequalities matter because many real constraints are not exact: budgets are “at most,” heights are “at least,” and time limits are “less than.” On the PSAT 8/9, you’ll solve inequalities and interpret solution sets on number lines (one variable) and shaded regions (two variables). ### Solving linear inequalities in one variable Solving is similar to equations: isolate the variable using inverse operations. The big extra rule is about multiplying or dividing by a negative. **Critical rule**: If you multiply or divide both sides by a negative number, you must **flip the inequality sign**. Example 1 (no sign flip needed): Solve: 2x + 5 \le 17 Subtract 5: 2x \le 12 Divide by 2: x \le 6 Example 2 (sign flips): Solve: -3x + 4 > 10 Subtract 4: -3x > 6 Divide by -3$and flip$>$to$

4. Shade the side that makes the inequality true.

Example:
$y > -x + 2$

• Boundary line: $y = -x + 2$ (dashed because $>$).
• Test point \,(0, 0)\,$: check 0 > -(0) + 2 0 > 2 False, so shade the side **not** containing \,(0, 0)\,$.

Compound inequalities (between two numbers)

Sometimes you see:
$-1 \le 2x + 3 < 7$
This means both are true at once. Solve it like a system:

Subtract $3$ everywhere:
$-4 \le 2x < 4$
Divide by $2$:
$-2 \le x < 2$

Inequalities as constraints in real problems

If a movie ticket costs $\$12$ and you have at most $\$50$, and you want to buy $t$ tickets:
$12t \le 50$
Solving gives a maximum value of $t$, but you also must interpret that $t$ should be a whole number (you can’t buy $\frac{25}{6}$ tickets). That extra interpretation step is frequently tested.

Exam Focus

• Typical question patterns:
  • Solve one-variable inequalities, including those requiring a sign flip.
  • Graph solution sets on a number line or match a graph to an inequality.
  • Graph a two-variable inequality by shading the correct region.
• Common mistakes:
  • Forgetting to flip the inequality sign when dividing or multiplying by a negative.
  • Using a solid line when the inequality is strict (or dashed when it includes equality).
  • Testing a point that lies on the boundary line (which can confuse the true/false check).