New Jersey Grade 8 Math

A system’s solution must satisfy both equations at the same time. The 2023 New Jersey standard includes solving linear systems by substitution and estimating solutions from graphs. This original example solves a two-equation system first, then uses a table to check the result.

Try this original activity

A classroom printing model has two invented offers for identical display sheets. A costs $8 to set up and $3 per sheet. B costs $2 to set up and $5 per sheet. Write equations for total cost C and number of sheets n. Use substitution to find when the costs agree. Assume nonnegative whole-sheet quantities and no other charges.

Work through the reasoning

The equations are C = 3n + 8 and C = 5n + 2. Substitute the first expression for C into the second equation: 3n + 8 = 5n + 2. Subtract 3n and then subtract 2 to obtain 6 = 2n, so n = 3. Substitute n = 3 into either equation to find C = 17. Check both: 3 × 3 + 8 = 17 and 5 × 3 + 2 = 17. The system’s solution is (n, C) = (3, 17), meaning three sheets cost seventeen dollars under either model. At two sheets A costs fourteen dollars and B twelve; at four sheets A costs twenty and B twenty-two. Those nearby values confirm that the equality occurs between the two different cost orderings. A graph would represent the solution by the intersection (3, 17), because that point belongs to both lines.

Check the common mistake

Finding n = 3 does not mean the shared total cost is three dollars. Interpret both coordinates with their units. Comparing only the per-sheet charge leaves out the starting charges and can select the wrong cheaper offer for a small order. A table can confirm this solution, but the substitution calculation explains exactly where equality comes from. The classroom offers are invented and are not current supplier prices. This activity models one linear system and does not cover every kind of system or all of Grade 8 algebra.

Try a related case

Change A’s setup charge to $9 while keeping both per-sheet rates and B’s setup charge. Substitution gives 3n + 9 = 5n + 2, hence n = 3.5. The extended lines intersect, but the stated whole-sheet purchasing model has no order with equal cost. Explain the difference between a mathematical intersection and an allowed quantity in the situation.

Selected 2023 New Jersey references for this activity

The 2023 New Jersey Student Learning Standards for Mathematics define grade-specific expectations. These selected references point to the activity’s focus; read the complete official Grade 8 page, its examples and clarifications, and the mathematical practices alongside the school’s teaching sequence.

Sources checked October 6, 2026. These focus labels are independent paraphrases of selected expectations, not a complete state checklist or certified alignment of the lesson library.

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.

The official 2026–2027 schedule lists NJSLA-A mathematics for Grades 3 through high school, including Grade 8. NJDOE describes the mathematics assessment as adaptive. Confirm the learner’s assigned pathway, tools, and school 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 official adaptive assessment overview.

Read the 2026–2027 statewide assessment schedule and confirm school directions. 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.