Louisiana Grade 8 Math

How does simplifying reveal the solution set?

Compare these original equations: 4(x − 3) = 4x − 12; 4(x − 3) = 4x − 10; and 4(x − 3) = 2x + 2. They share the same left side but make different claims about equality. Simplify each by equivalent transformations and explain whether it has infinitely many real solutions, no real solution, or exactly one.

Start with the distributive property

In all three equations, 4(x − 3) becomes 4x − 12. The factor four applies to both terms. In the first equation, subtracting 4x from both sides leaves −12 = −12. That statement is true regardless of x, so every real value satisfies the original equation. It does not mean x equals negative twelve. The variable terms cancel because the two original expressions are identical.

Recognize a contradiction

In the second equation, subtracting 4x from both sides leaves −12 = −10. That is false for every x. No choice of x can repair a difference between these constant terms after equivalent transformations have removed the variable. For instance, setting x to zero gives negative twelve on the left and negative ten on the right, but one failed trial alone would not prove no solution. The contradiction obtained algebraically establishes that all choices fail.

Solve and check the third equation

The third simplifies to 4x − 12 = 2x + 2. Subtract 2x from both sides and add twelve to both sides to get 2x = 14. Divide both sides by two: x = 7. Substitution into the original equation gives 4(7 − 3) = 16 on the left and 2(7) + 2 = 16 on the right. The equivalent steps lead to one specified value, and substitution checks it in the original form. Keep the solution count and the solution value distinct in the explanation.

Explain why cancellation is not a complete answer

In both the first and second equations, the variable terms disappear. Their outcomes differ because the remaining constant statements differ: one is always true, the other always false. “The x terms cancel” is therefore an observation, not a final classification. If a student distributes only to x and writes 4x − 3, revisit the meaning of multiplying the entire group. Then try a new set of teacher-selected equations with the same three possible outcomes. Connect each classification with its equivalent final form rather than memorizing a label from this example.

Selected references from Louisiana mathematics standards

The Louisiana mathematics standards directory lists current and updated standards. For 2026–27, use the current K–12 document, updated December 21, 2017; the revised standards have full implementation in 2027–28. These selected references identify the worked activity’s focus. The expectations include numbered requirements, examples, and mathematical practices. Read the full document and confirm the school’s current instructional sequence.

Sources checked October 6, 2026. Read complete requirements, clarifications, limitations, and mathematical practices alongside the school’s sequence. This selected activity is an entry point for instruction.

Connect the explanation with practice

Use the lesson, worksheet, and quiz directory to choose a related skill. The existing resource links remain available; open each activity’s directions to understand its timer, tools, question count, and scoring.

Grade 8 is within Louisiana’s general LEAP mathematics testing range. The linked learning resources have their own directions and scoring; a practice score does not establish an official state performance level. Read Louisiana’s assessment overview.

Keep one example of the student’s reasoning with the practice result. Choose a smaller model, a related question, or a new context according to the difficulty revealed in the worked activity.