When should you use substitution versus elimination?

Use substitution when one equation is already solved for a variable, or is easy to solve, then substitute that expression into the other equation. Use elimination when a variable has matching or opposite coefficients, or can be made so by multiplying, then add or subtract to cancel it. Both methods give the same ordered-pair solution, so pick whichever the system makes easier.

What does the solution to a system of equations represent?

It is the ordered pair (x, y) that satisfies both equations at once, the point where the two lines intersect. One crossing point gives one solution. Parallel lines (same slope, different intercept) never cross and give no solution. Identical lines overlap completely and give infinitely many solutions.

How do you decide whether a boundary line is solid or dashed?

Use a solid line for the inequalities that include equality, the less-than-or-equal and greater-than-or-equal signs, because points on the line are solutions. Use a dashed line for the strict less-than and greater-than signs, because points on the line are not solutions. This is the two-dimensional version of the open and closed circles on a number line.

Which side of a boundary line do you shade?

Pick a test point that is not on the line, often the origin (0, 0), and substitute it into the inequality. If the inequality is true at that point, shade the side containing it. If it is false, shade the other side. A single test point fixes the entire half-plane, because the truth value can only change across the boundary.

What does the overlapping region of a system of inequalities mean?

Each inequality shades a half-plane, and the solution to the system is where all the half-planes overlap, the set of points that satisfy every inequality at once. A point solves the system only if it makes all the inequalities true. In modeling, this overlap is called the feasible region, the set of all viable options.

How do you set up a system from a word problem?

Define two variables, then write one equation or inequality for each condition. A common pattern is one equation for a count (total number) and one for a value (total money or weight). Limits like at most and at least become inequalities, and quantities cannot be negative. Solve, then check that the answer is viable, whole numbers for counts and nonnegative amounts.

LouisianaMaths

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

A deep-dive Louisiana LEAP 2025 Algebra I guide to systems of equations and inequalities: solving systems by substitution and elimination (A-REI.C.6), solving by graphing (A-REI.D.11), graphing a two-variable inequality and a system of inequalities (A-REI.D.12), and modeling with constraints (A-CED.A.3).

Generated by Claude Opus 4.814 min readA1: A-REI.C.6, A-REI.D.12, A-CED.A.3Updated 2026-06-02

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

Jump to a section

What this module covers
Solving systems algebraically
Solving by graphing
Graphing a linear inequality
Systems of inequalities
Modeling with constraints
How this module is examined
Check your knowledge

What this module covers

This guide covers systems of equations and inequalities on the Louisiana LEAP 2025 Algebra I test, part of the Major Content category: solving systems algebraically with substitution and elimination (A-REI.C.6), solving by graphing (A-REI.D.11), graphing a two-variable inequality and a system of inequalities (A-REI.D.12), and modeling with constraints (A-CED.A.3). Each dot-point page has its own practice: solving systems algebraically, solving systems by graphing, graphing linear inequalities, systems of linear inequalities, and modeling with systems.

Solving systems algebraically

A system is two equations that must both hold; the solution is the ordered pair where the lines meet.

If both variables cancel with a true statement, the lines coincide (infinitely many solutions); a false statement means parallel lines (no solution).

Solving by graphing

Graph both lines; the intersection is the solution. The number of crossings is the number of solutions: one (different slopes), none (parallel), or infinitely many (identical lines).

Graphing a linear inequality

A two-variable inequality graphs as a half-plane: graph the boundary (solid for $\le$ or $\ge$ , dashed for $<$ or $>$ ), then shade the side a test point satisfies.

Systems of inequalities

Graph each inequality, then take the overlap: the region satisfying all of them. A point solves the system only if it makes every inequality true. In modeling, this overlap is the feasible region.

Modeling with constraints

Define two variables and write one equation or inequality per condition (often a count equation and a value equation). Limits become inequalities, and quantities are nonnegative. Solve, then judge viability: counts must be whole, amounts nonnegative.

How this module is examined

Equation response. Solve a system and enter the ordered pair, or solve one from a word problem.
Graphing items. Plot lines and intersections; graph half-planes and overlapping regions.
Type III modeling. Set up and solve a system from a context, then interpret viability.
Multiple choice. Identify the number of solutions, the correct inequality, or the solution region.

Check your knowledge

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

Solve $y = 2x - 1$ and $3x + y = 14$ . (3 points)
Solve $2x + 3y = 12$ and $2x - y = 4$ by elimination. (3 points)
A system graphs as two parallel lines. How many solutions? (1 point)
Two lines cross at $(2, 3)$ . What is the solution? (1 point)
When graphing $y > 2x + 1$ , is the boundary solid or dashed, and which side is shaded? (2 points)
Is $(1, 1)$ a solution to $2x + y \le 4$ ? (2 points)
Is $(2, 1)$ a solution to $y \le x + 2$ and $y > -x$ ? (2 points)
Describe the solution set of a system of two linear inequalities. (1 point)
A store sells 200 tickets ( $8 adult,$ 5 child) for $1360 total. How many adult tickets? (4 points)
Model "at most 40 items" for $x$ chairs and $y$ tables. (1 point)

Solutions to check your knowledge

Use these worked steps to check your method before comparing answers.

Step 1: Q1 - substitution

The first equation is already solved for $y$ , so substitute $y = 2x - 1$ directly into the second equation. Once you have $x$ , substitute back to find $y$ .

Substituting: $3x + (2x - 1) = 14 \Rightarrow 5x - 1 = 14 \Rightarrow 5x = 15 \Rightarrow x = 3$ . Then $y = 2(3) - 1 = 5$ .

Answer: $(3, 5)$ .

Step 2: Q2 - elimination

Both equations contain $2x$ , so subtracting one from the other cancels the $x$ terms, leaving a single equation in $y$ . Then substitute $y$ back to find $x$ .

Subtracting the second from the first: $(2x + 3y) - (2x - y) = 12 - 4$ , giving $4y = 8$ , so $y = 2$ . Then $2x - 2 = 4$ , giving $x = 3$ .

Answer: $(3, 2)$ .

Step 3: Q3 - parallel lines

Parallel lines have the same slope but different y-intercepts. They never intersect, so no point satisfies both equations at once.

Answer: No solution.

Step 4: Q4 - reading the intersection

The solution to a system is the ordered pair where both lines meet. If the graph shows the crossing point, that point is the answer directly.

Answer: $(2, 3)$ .

Step 5: Q5 - boundary and shading for a strict inequality

The strict greater-than sign means the boundary line itself is not part of the solution, so draw it dashed. Then test a point not on the line. Substituting $(0, 0)$ : $0 > 2(0) + 1 = 1$ is false, so shade the side that does not contain the origin, which is above the line.

Answer: Dashed boundary; shade above.

Step 6: Q6 - testing a point in an inequality

Substitute $x = 1$ and $y = 1$ into $2x + y \le 4$ . A true result means the point is in the solution region.

$2(1) + 1 = 3 \le 4$ is true.

Answer: Yes, $(1, 1)$ is a solution.

Step 7: Q7 - testing a point in a system of inequalities

A point solves a system only if it satisfies every inequality. Test $(2, 1)$ in each one.

$y \le x + 2$ : $1 \le 2 + 2 = 4$ , true. $y > -x$ : $1 > -2$ , true. Both hold.

Answer: Yes, $(2, 1)$ is a solution to the system.

Step 8: Q8 - solution set of a system of inequalities

Each inequality shades a half-plane. A point solves the system only if it satisfies every inequality, meaning it lies in every shaded region at once.

Answer: The region where all shaded half-planes overlap.

Step 9: Q9 - word problem with a system

Define two variables, write one equation for the total count and one for the total revenue, then solve. Let $a$ = adult tickets and $c$ = child tickets.

Count: $a + c = 200$ . Revenue: $8a + 5c = 1360$ .

From the count equation, $c = 200 - a$ . Substitute: $8a + 5(200 - a) = 1360$ , giving $8a + 1000 - 5a = 1360$ , so $3a = 360$ and $a = 120$ .

Answer: $120$ adult tickets.

Step 10: Q10 - modeling a constraint with an inequality

"At most 40 items" means the total number of chairs and tables combined cannot exceed 40. Use the less-than-or-equal symbol because equality (exactly 40) is still allowed.

Answer: $x + y \le 40$ .

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)