Montana Grade 8 Math

Solving a linear equation can produce a value, a statement true for every value, or a contradiction. Use equivalent transformations to decide which case applies. The three original equations below differ by one important term.

Try this original activity

Classify each equation over the real numbers: A: 4(x + 2) = 4x + 8; B: 4(x + 2) = 4x + 5; C: 4(x + 2) = 3x + 8. Show the simplification that supports each classification.

Work through the reasoning

Expand the left side to 4x + 8 in all three equations. For A, subtracting 4x gives 8 = 8, a true statement independent of x. Every real value is a solution, so there are infinitely many solutions. For B, the same subtraction gives 8 = 5, which is false. No value can satisfy it, so there is no solution. For C, subtracting 3x gives x + 8 = 8, and subtracting 8 gives x = 0. Substitution checks C: both sides equal 8 at zero. The decisive difference is whether the variable terms cancel and whether the remaining constants agree.

Check the common mistake

When x disappears, do not automatically report x = 0. A gives a true identity, while B gives a contradiction. Trying one value can disprove that particular value as a solution, but cannot alone prove that every value fails. The equivalent transformations in B supply the general reason. Also distribute the 4 to both x and 2; writing 4x + 2 changes the equation. All operations here preserve equality when applied to both sides.

Try a related case

Classify 2(y − 3) = 2y − 6, 2(y − 3) = 2y − 1, and 2(y − 3) = y − 6. They produce an identity, a contradiction, and y = 0 respectively. Explain the number of solutions before checking selected values.

Selected references from Montana mathematics standards

Montana’s Office of Public Instruction identifies July 1, 2026 as the implementation date for its revised mathematics standards. The selected references below were checked in the current PK–12 mathematics document linked from the OPI mathematics directory. Use the 2026 implementation document when checking a code; earlier Montana documents use a different set of references. 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.

OPI identifies MAST English language arts and mathematics assessment for Grades 3–8, including Grade 8. The required-assessments directory also identifies alternate assessment pathways. Follow the school’s current assigned pathway, instructional sequence, and administration directions. Independently published practice may use its own timer, tools, question count, and scoring; these are not official administration specifications or a prediction of a state performance level. Read the current OPI assessment overview.

Read OPI mathematics guidance for later guidance. 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.