PSAT 10 Math Algebra: Linear Relationships, Systems, and Inequalities

Linear equations in one variable

A linear equation in one variable is an equation where the variable (usually $x$) appears only to the first power and is not multiplied by itself or inside a function like a square root. “Linear” means the variable changes at a constant rate—no curves, no exponents like $x^2$.

You can think of a linear equation as a balance scale: the left side and right side must stay equal. Solving the equation is the process of performing the same operation on both sides to keep the balance while isolating the variable.

What linear equations look like

Most PSAT linear equations in one variable can be written (or rearranged) into the form:

$ax + b = c$

where:

• $x$ is the variable you’re solving for
• $a$ is the coefficient (the number multiplying $x$)
• $b$ and $c$ are constants (plain numbers)

Sometimes the equation is already “simple,” and sometimes you must simplify first by combining like terms or distributing.

Why this matters

Linear equations are the foundation for almost everything else in Algebra:

• Solving inequalities uses the same steps (with one extra rule).
• Systems of equations rely on solving linear equations efficiently.
• Linear functions are basically equations you interpret as input-output rules.

If you can reliably isolate a variable, you can handle a large portion of PSAT algebra questions.

How solving works (the core moves)

To solve linear equations, you use inverse operations to undo what’s happening to $x$.

Common moves, in a typical order:

1. Simplify each side: distribute, combine like terms.
2. Move variable terms to one side: add or subtract the same expression on both sides.
3. Move constant terms to the other side.
4. Divide (or multiply) to make the coefficient of $x$ equal to $1$.

A good habit is to keep your work in clear steps—errors often come from skipping steps and dropping negatives.

Example 1: Solve a basic linear equation

Solve:

$3x - 7 = 11$

Step 1: Add $7$ to both sides (undo subtracting $7$):

$3x = 18$

Step 2: Divide both sides by $3$ (undo multiplying by $3$):

$x = 6$

So the solution is:

$x = 6$

Example 2: Solve with distribution and combining like terms

Solve:

$2(3x - 4) + x = 5$

Step 1: Distribute $2$:

$6x - 8 + x = 5$

Step 2: Combine like terms on the left:

$7x - 8 = 5$

Step 3: Add $8$ to both sides:

$7x = 13$

Step 4: Divide by $7$:

$x = \frac{13}{7}$

Special cases: no solution or infinitely many solutions

Sometimes, after simplifying, the variable disappears.

• No solution happens when you get a false statement, like:

$0 = 5$

This means there is no value of $x$ that can make the original equation true.

• Infinitely many solutions happens when you get a true statement, like:

$0 = 0$

This means every value of $x$ satisfies the equation (the two sides were really the same expression).

A common PSAT framing is: “How many solutions does the equation have?” or “For what value of $k$ does the equation have no solution?” Those problems test whether you can recognize when two linear expressions are identical or parallel-shifted versions of each other.

Exam Focus

• Typical question patterns:
  • Solve for $x$ after distributing and combining like terms.
  • Determine whether an equation has one solution, no solution, or infinitely many solutions.
  • Solve for a parameter (like $k$) so that the equation has a specific number of solutions.
• Common mistakes:
  • Dropping a negative when moving terms (remember you add/subtract the entire term).
  • Distributing incorrectly, especially with subtraction: $-(x-3)$ should become $-x+3$.
  • Dividing only one term by a number instead of the entire side.

Linear equations in two variables

A linear equation in two variables involves two variables (often $x$ and $y$) to the first power. The graph of such an equation is always a straight line in the coordinate plane.

A common general form is:

$ax + by = c$

where $a$, $b$, and $c$ are constants, and $a$ and $b$ are not both zero.

Why this matters

Two-variable linear equations are the language of straight-line relationships. On the PSAT, they show up when you:

• Interpret a line from a graph or table.
• Convert between equation forms.
• Model a real situation like cost vs. number of items.

Even when a question doesn’t say “graph this,” you’re often expected to think about what the line would look like.

Common forms and what they tell you

Different forms highlight different features of the same line.

Here:

• Slope $m$ is the rate of change (how much $y$ changes when $x$ increases by 1).
• $y$-intercept $b$ is the value of $y$ when $x = 0$.

How to graph a two-variable linear equation

There are a few reliable strategies:

1. Use slope-intercept form $y = mx + b$:

   • Plot $b$ on the $y$-axis.
   • Use slope $m = \frac{\text{rise}}{\text{run}}$ to get a second point.
   • Draw the line.

2. Find two points:

   • Choose convenient $x$ values and compute $y$.
   • Or find intercepts by setting $x = 0$ and $y = 0$.

3. Convert standard form to slope-intercept:

   • Solve for $y$.

Example 1: Convert standard form to slope-intercept form

Rewrite:

$3x + 2y = 8$

Solve for $y$:

$2y = -3x + 8$

$y = -\frac{3}{2}x + 4$

Now you can read:

• slope $m = -\frac{3}{2}$
• $y$-intercept $b = 4$

Example 2: Find intercepts from standard form

Find the intercepts of:

$4x + 5y = 20$

• $x$-intercept: set $y = 0$:

$4x = 20$

$x = 5$

So the $x$-intercept is $\left(5,0\right)$.

• $y$-intercept: set $x = 0$:

$5y = 20$

$y = 4$

So the $y$-intercept is $\left(0,4\right)$.

A common misconception is to think the constant $20$ is an intercept by itself. Intercepts are coordinate points, found by setting the other variable to zero.

Exam Focus

• Typical question patterns:
  • Convert between forms (especially standard to slope-intercept).
  • Identify slope and intercepts from an equation.
  • Match an equation to a graph (or vice versa) by using slope and intercept.
• Common mistakes:
  • Sign errors when solving for $y$ (watch the move of $Ax$ to the other side).
  • Confusing slope with $y$-intercept.
  • Treating $Ax + By = C$ as if $A$ and $B$ were the intercepts.

Linear functions

A function is a rule that assigns each input exactly one output. A linear function is a function whose outputs change at a constant rate, so its graph is a line.

A linear function is often written as:

$f(x) = mx + b$

This is the same structure as $y = mx + b$—it’s just function notation. Here:

• $f(x)$ means “the output when the input is $x$.”
• $m$ is the slope (rate of change).
• $b$ is the initial value (output when input is $0$).

Why this matters

PSAT questions often describe a situation (like costs, distances, or points scored) and expect you to:

• Identify the rate of change and starting value.
• Build an equation or function.
• Interpret what the slope and intercept mean in context.

Understanding linear functions is how you turn words, tables, and graphs into algebra.

Slope as a rate of change

Slope is fundamentally:

$m = \frac{\Delta y}{\Delta x}$

That fraction describes “change in output per change in input.”

• If $m > 0$, the function increases as $x$ increases.
• If $m < 0$, it decreases.
• If $m = 0$, it’s constant: $f(x) = b$.

In real life, slope might mean dollars per item, miles per hour, points per game, etc.

Writing a linear function from information

There are a few standard ways PSAT problems give you enough information.

1) From slope and a point

If you know slope $m$ and a point $\left(x_1, y_1\right)$ on the line, use point-slope form:

$y - y_1 = m(x - x_1)$

Then convert to $y = mx + b$ if needed.

Example 1: Write the equation given a slope and point
A line has slope $2$ and passes through $\left(3,-1\right)$. Find its equation.

Start with point-slope form:

$y - (-1) = 2(x - 3)$

$y + 1 = 2x - 6$

$y = 2x - 7$

So the linear function could be written as:

$f(x) = 2x - 7$

2) From two points

If you know two points, find slope first:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Then use point-slope form.

Example 2: Find a linear function through two points
Find the equation of the line through $\left(2,5\right)$ and $\left(6,1\right)$.

Compute slope:

$m = \frac{1 - 5}{6 - 2} = \frac{-4}{4} = -1$

Use point-slope with $\left(2,5\right)$:

$y - 5 = -1(x - 2)$

$y - 5 = -x + 2$

$y = -x + 7$

So:

$f(x) = -x + 7$

A common mistake is flipping the slope fraction inconsistently. If you do $\frac{y_2-y_1}{x_2-x_1}$, keep the same order in both numerator and denominator.

Function notation vs. equation notation

Students sometimes treat $f(x)$ as multiplication, like $f \times x$. It’s not. It’s a single symbol meaning “output of the function.”

For example, if $f(x) = 3x + 2$, then:

$f(4) = 3(4) + 2 = 14$

That substitution skill appears constantly in PSAT problems.

Exam Focus

• Typical question patterns:
  • Interpret $m$ and $b$ in a real context (rate and initial value).
  • Write a linear function from two points, a table, or a graph.
  • Evaluate a function at a given input, like $f(7)$.
• Common mistakes:
  • Confusing $f(x)$ with $f \times x$ or trying to “solve for $f$.”
  • Mixing up slope and intercept when building a model from words.
  • Using the wrong order when computing slope from two points.

Systems of two linear equations in two variables

A system of linear equations is a set of two (or more) equations that share the same variables. For a system with two linear equations in two variables, you’re looking for the point $\left(x,y\right)$ that satisfies both equations at the same time.

Graphically, each equation is a line. Solving the system means finding where the lines intersect.

Why this matters

Systems are a powerful modeling tool: many real situations involve two conditions happening simultaneously (budget and quantity, two pricing plans, mixtures, etc.). On the PSAT, systems questions often test whether you can:

• Find the intersection point.
• Decide how many solutions exist.
• Interpret the solution in context.

Number of solutions (what the graph tells you)

Two lines in a plane can relate in three ways:

1. One solution: lines intersect once (different slopes).
2. No solution: lines are parallel (same slope, different intercepts).
3. Infinitely many solutions: lines are the same line (equations are equivalent).

This connects to the “special cases” you saw with one-variable equations: sometimes simplification reveals a contradiction (no solution) or an identity (infinitely many).

Methods to solve

You’re typically expected to use either substitution or elimination. Both are valid; choose what looks easiest.

Substitution

Use substitution when one equation is already solved for a variable (or can be easily).

How it works:

1. Solve one equation for $x$ or $y$.
2. Substitute that expression into the other equation.
3. Solve the resulting one-variable equation.
4. Plug back in to find the other variable.

Example 1: Solve by substitution

$y = 2x + 1$

$3x + y = 16$

Substitute $y = 2x + 1$ into the second equation:

$3x + (2x + 1) = 16$

$5x + 1 = 16$

$5x = 15$

$x = 3$

Now plug into $y = 2x + 1$:

$y = 2(3) + 1 = 7$

Solution:

$\left(3,7\right)$

A common mistake is to substitute into the wrong place or forget parentheses, especially when substituting a negative expression.

Elimination (also called linear combination)

Use elimination when the equations line up nicely so one variable can cancel.

How it works:

1. If needed, multiply one or both equations so coefficients match.
2. Add or subtract the equations to eliminate one variable.
3. Solve for the remaining variable.
4. Substitute back.

Example 2: Solve by elimination

$2x + 3y = 13$

$4x - 3y = 5$

Add the equations (because $+3y$ and $-3y$ cancel):

$6x = 18$

$x = 3$

Substitute into the first equation:

$2(3) + 3y = 13$

$6 + 3y = 13$

$3y = 7$

$y = \frac{7}{3}$

Solution:

$\left(3,\frac{7}{3}\right)$

A very common error is combining equations but forgetting to apply the operation to every term on both sides.

Systems in context

If a problem says two plans cost the same at a certain number of months, that “same cost” is an intersection of two lines (same $y$ value at the same $x$). In these contexts:

• $x$ often represents time/quantity.
• $y$ represents total cost/value.
• The solution point gives both the quantity and the matching cost.

Exam Focus

• Typical question patterns:
  • Solve a system and interpret $x$ and $y$ in a word problem.
  • Determine the number of solutions by comparing slopes/intercepts or by simplifying.
  • Choose which method (substitution vs elimination) is most efficient.
• Common mistakes:
  • Arithmetic sign errors when adding/subtracting equations.
  • Solving for one variable but forgetting to plug back in to get the ordered pair.
  • Mixing up the meaning of $x$ and $y$ in a real-world interpretation.

Linear inequalities in one or two variables

A linear inequality looks like a linear equation, but instead of an equals sign, it uses:

$,\;\le,\;\ge$

Solving an inequality means finding all values that make it true. The result is not usually a single number—it’s a set of numbers (often an interval) or a region on a graph.

Why this matters

Inequalities model constraints—limits, minimums, maximums, and ranges. On the PSAT, you’ll see inequalities when problems talk about:

• at least / at most
• no more than / no less than
• must be greater than / must be below

They also appear in graph interpretation questions and “which values satisfy” multiple conditions.

Solving linear inequalities in one variable

The process is almost identical to solving equations, with one critical rule:

If you multiply or divide both sides by a negative number, you must flip the inequality sign.

That rule exists because multiplying by a negative reverses order on the number line.

Example 1: Solve a one-variable inequality

Solve:

$5 - 2x \ge 11$

Subtract $5$ from both sides:

$-2x \ge 6$

Divide by $-2$ and flip the sign:

$x \le -3$

So the solution set is all $x$ values less than or equal to $-3$.

A common mistake is forgetting to flip the inequality when dividing by a negative.

Compound inequalities

Sometimes you’ll see an inequality like:

$2 < x + 1 \le 7$

You solve it by treating it like two inequalities at once.

Subtract $1$ from all three parts:

$1 < x \le 6$

This means $x$ is greater than $1$ and less than or equal to $6$.

Inequalities in two variables (graphing half-planes)

A linear inequality in two variables describes a region of the coordinate plane rather than a single line.

A typical form is:

$y > mx + b$

To graph it:

1. Graph the boundary line $y = mx + b$.
2. Decide whether the boundary is included.
3. Shade the side that satisfies the inequality.

Solid vs. dashed boundary

• If the inequality is $\le$ or $\ge$, the boundary line is solid (points on the line are included).
• If the inequality is $<$ or $>$, the boundary line is dashed (points on the line are not included).

How to choose which side to shade

After drawing the boundary line, you can use a test point—a point not on the line—to see which side works. The easiest test point is often $\left(0,0\right)$ (unless it lies on the boundary line).

Example 2: Graph and interpret a two-variable inequality

Graph the solution set of:

$y \ge -\frac{1}{2}x + 3$

• Boundary line: $y = -\frac{1}{2}x + 3$.
• Because it’s $\ge$, draw a solid line.
• Shade the region above the line, because $y$ is greater than or equal to the line’s $y$-values.

If you want to confirm shading with a test point, try $\left(0,0\right)$:

$0 \ge -\frac{1}{2}(0) + 3$

$0 \ge 3$

This is false, so $\left(0,0\right)$ is not in the solution region. That means you shade the side of the line that does not contain the origin.

Systems of linear inequalities

A system of inequalities represents the overlap of multiple regions. The solution is the set of points that satisfy all inequalities at once (the intersection of shaded regions).

On the PSAT, you might not have to fully draw a complex region, but you may be asked which points satisfy a system, or which graph matches a system.

A practical way to check a point against a system is to substitute the point into each inequality and confirm all are true.

Exam Focus

• Typical question patterns:
  • Solve a one-variable inequality and select the correct interval/number line.
  • Graph a two-variable inequality by drawing a boundary line and shading.
  • Test whether a point satisfies an inequality or a system of inequalities.
• Common mistakes:
  • Forgetting to flip the inequality when multiplying/dividing by a negative.
  • Using a solid line for $<$ or $>$ (or dashed for $\le$ or $\ge$).
  • Shading the wrong side because slope-intercept form is misread or the test point check is skipped.