What are the three solution cases for a linear equation?

After simplifying, if the variable remains you get one solution. If the variable cancels and a true statement remains (like 5 equals 5), there are infinitely many solutions, an identity. If the variable cancels and a false statement remains (like 5 equals 2), there is no solution. You decide by simplifying until the variable disappears, then reading the leftover statement.

Why does the inequality sign flip when you multiply or divide by a negative?

Multiplying by a negative reflects every number across zero, which reverses their order on the number line. Starting from a true statement like 3 is less than 5, multiplying by negative one gives negative 3 and negative 5, and negative 3 is greater than negative 5. To keep the statement true you must reverse the inequality. Adding or subtracting never does this; only a negative multiply or divide.

How do you solve a formula for a different variable?

Treat every other letter as a constant and isolate the target variable with the properties of equality, undoing operations in reverse order. For the triangle area A equals one-half base times height, solving for height gives height equals 2A over base. The answer is an expression in the other letters, not a single number, and there may be more than one correct equivalent form.

What do slope and the y-intercept mean in a real-world model?

In y equals mx plus b, the slope m is the rate of change (dollars per week, miles per hour) and b is the starting value, the amount when x is zero. For savings of y equals 5x plus 30, the slope 5 means saving 5 dollars per week and the intercept 30 is the starting savings. Always state the units when you interpret them.

How do you write the equation of a line from two points?

Find the slope first using rise over run, then use point-slope form y minus y-one equals m times (x minus x-one) with either point, and rearrange to slope-intercept form if asked. If one of the points has x equal to zero, that point is the y-intercept, so you can read b directly and skip a step.

How do you translate a word problem into an inequality?

Define the variable first, then translate the words: a fixed amount is a constant, a per-unit amount is a rate that multiplies the variable, and the comparison word picks the sign. At most or no more than is less-than-or-equal, at least is greater-than-or-equal, more than is strictly greater. Build it, solve it, and interpret the answer in context with units.

LouisianaMaths

Louisiana LEAP 2025 Algebra I: a complete guide to linear equations and inequalities (A-REI, A-CED, F-IF)

A deep-dive Louisiana LEAP 2025 Algebra I guide to linear equations and inequalities: solving one-variable equations and inequalities (A-REI.B.3), rearranging formulas (A-CED.A.4), creating models from context (A-CED.A.1), and slope, intercepts, and writing equations of lines (F-IF.B).

Generated by Claude Opus 4.815 min readA1: A-REI.B, A-CED.A, F-IF.BUpdated 2026-06-02

Reviewed by: AI editorial process; not yet individually human-reviewed

Jump to a section

What this module covers
Solving linear equations
Rearranging formulas
Solving inequalities
Creating models from context
Slope, intercepts, and writing lines
How this module is examined
Check your knowledge

What this module covers

This guide covers linear equations and inequalities on the Louisiana LEAP 2025 Algebra I test, the heart of the Major Content category: solving one-variable equations and inequalities (A-REI.B.3), rearranging formulas (A-CED.A.4), creating models from a context (A-CED.A.1), and slope, intercepts, and writing equations of lines (F-IF.B). Several of these are no-calculator fluency skills on Session 1a. Each dot-point page has its own practice: solving linear equations, literal equations and formulas, solving linear inequalities, creating equations and inequalities, linear functions and slope, and writing equations of lines.

Solving linear equations

Isolate the variable with the properties of equality, doing the same operation to both sides. Clear parentheses, clear fractions, gather variables and constants, then divide. If the variable cancels with a true statement, there are infinitely many solutions; a false statement means no solution.

Rearranging formulas

Treat the other letters as constants and isolate the chosen variable, undoing operations in reverse order. From $A = \tfrac{1}{2}bh$ , solving for $h$ gives $h = \frac{2A}{b}$ . The result is a general template that works for every value at once.

Solving inequalities

Solve like an equation, with one extra rule: a negative multiply or divide flips the sign.

Creating models from context

Define the variable, translate the words (constant for a fixed amount, rate for a per-unit amount, comparison word for the sign), solve, and interpret.

Slope, intercepts, and writing lines

Slope is $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}$ , the rate of change. In $y = mx + b$ , $m$ is the slope and $b$ the $y$ -intercept. Set $y = 0$ for the $x$ -intercept. To write a line, use point-slope $y - y_1 = m(x - x_1)$ from a slope and point, finding the slope first if you have two points.

How this module is examined

Equation response. Solve an equation or inequality, rearrange a formula, or compute a slope or intercept.
Type III modeling. Build and solve a model from a described situation, then interpret with units.
Type II reasoning. Justify the number of solutions or interpret a slope.
Graphing items. Number-line solutions for inequalities; lines and intercepts on the coordinate plane.

Check your knowledge

Work these as you would for credit on the online test.

Solve $4(x - 3) = 2x + 6$ . (2 points)
How many solutions does $3x + 5 = 3x - 2$ have? (1 point)
Solve $A = \tfrac{1}{2}bh$ for $h$ . (2 points)
Solve $-3x + 4 \le 19$ . (2 points)
How is $x > 3$ graphed on a number line? (1 point)
A gym charges $40 plus$ 25 per month. Write $C$ in terms of months $m$ , then find $m$ when $C = 215$ . (3 points)
A student has $60 and tickets cost$ 4.50. Write an inequality for the number of tickets $t$ . (2 points)
Find the slope through $(1, 2)$ and $(4, 11)$ . (2 points)
Find the $x$ -intercept of $y = 2x - 6$ . (2 points)
Write the line with slope $-2$ through $(3, 4)$ in slope-intercept form. (2 points)

Solutions

Step 1: Q1 - solving with variables on both sides

Collect all $x$ terms on one side by subtracting $2x$ , then solve for $x$ :

4x - 12 = 2x + 6 \implies 2x = 18 \implies x = 9

Step 2: Q2 - identifying no solution

When the variable cancels completely, check whether the remaining equation is true or false. A false statement means no value of $x$ works:

The $x$ terms cancel leaving $5 = -2$ (false): no solution.

Step 3: Q3 - solving a literal equation

Treat the variable $h$ like an unknown and isolate it by performing inverse operations on both sides:

2A = bh \implies h = \frac{2A}{b}

Step 4: Q4 - solving an inequality

Solve like an equation, but reverse the inequality sign whenever you multiply or divide both sides by a negative number:

-3x \le 15 \implies x \ge -5 \quad \text{(flip on dividing by } {-3}\text{)}

Step 5: Q5 - graphing an inequality

For $x > 3$ , use an open circle at $3$ (not included) and shade to the right (all values greater than $3$ ).

Step 6: Q6 - writing and solving a linear model

Identify the fixed cost and the rate of change from the context, write the equation, then substitute the given total to find the unknown:

$C = 40 + 25m$ ; $215 = 40 + 25m \implies 25m = 175 \implies m = 7$ .

Step 7: Q7 - inequality from a budget constraint

Let $t$ be the number of tickets and write the spending as an inequality. Solve and remember that tickets must be whole numbers:

$4.50t \le 60$ , so $t \le 13.33$ : at most 13 tickets.

Step 8: Q8 - slope from two points

Use the slope formula $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ with the two given points:

m = \frac{11 - 2}{4 - 1} = \frac{9}{3} = 3

Step 9: Q9 - x-intercept of a line

The $x$ -intercept is where the line crosses the $x$ -axis, where $y = 0$ . Set $y = 0$ and solve for $x$ :

0 = 2x - 6 \implies x = 3

Step 10: Q10 - point-slope form to slope-intercept form

Write the equation in point-slope form using the given point and slope, then rearrange to slope-intercept form:

y - 4 = -2(x - 3) \implies y = -2x + 6 + 4 \implies y = -2x + 10

Sources & how we know this

Louisiana Student Standards for Mathematics — Louisiana Department of Education (2025)
LEAP 2025 Assessment Guide for Algebra I — Louisiana Department of Education (2025)