Substitution in equations in 6th grade

Left-side values are $8$ and $11$ for $x = 3$ and $x = 6$. Only $x = 3$ makes the left side equal $8$, so $x = 3$ is the solution.

1. 1Substitution means replacing the variable with a number. For each value of x, replace x in $x + 5$ and compute the result.
2. 2With $x = 3$, the left side is $8$, which matches the right side — so $x = 3$ is the solution.
3. 3With $x = 6$, the left side is $11$.
4. 4The left-side values are $8$ and $11$. Only $x = 3$ makes the left side equal the right side ($8 = 8$), so $x = 3$ is the solution.

Common mistake. A common mistake is adding in the wrong order — for example, computing $6 + 5 = 11$ and then thinking the equation requires 11 on the right side, rather than checking which candidate gives exactly 8.

The idea underneath. To check whether a value is a solution, substitute it into the equation and evaluate each side separately.

The answer is 4 and 6 because those are the only numbers in the set that make the inequality true.

1. 1Start with the inequality.
2. 2Divide both sides by 3. That gives $x \geq 4$.
3. 3Check each candidate: 1, 2, and 3 are all less than 4, so they do not satisfy the inequality. The boundary value 4 makes 3(4) = 12, and 12 ≥ 12 is true, so 4 is included. The value 6 gives 3(6) = 18 ≥ 12, which is also true.
4. 4Write the solutions from least to greatest: 4, 6.

Common mistake. A common mistake is excluding 4. When x = 4, 3x = 12, and 12 ≥ 12 is true because the symbol is ≥, not >.

The idea underneath. A number is a solution only if substituting it makes the inequality true, and boundary values are included only for ≤ or ≥.

Marcus's left-side value is wrong at x = 3 (the correct value is $2(3 + 3) = 12$, not 11), so select the choice for x = 3.

1. 1To check Marcus's work, recompute the left side $2(x + 3)$ for each candidate value of x and compare it to what Marcus wrote.
2. 2For x = 1, the correct left side is $2(1 + 3) = 8$. Marcus wrote 8, so this row is correct.
3. 3For x = 2, the correct left side is $2(2 + 3) = 10$. Marcus wrote 10, so this row is correct.
4. 4For x = 3, the correct left side is $2(3 + 3) = 12$. Marcus wrote 11, so this row is wrong.
5. 5For x = 4, the correct left side is $2(4 + 3) = 14$. Marcus wrote 14, so this row is correct.
6. 6The left-side value is wrong at x = 3 (Marcus wrote 11, but $2(3 + 3) = 12$), so select the choice for x = 3.

Common mistake. Students sometimes select x-values that do not solve the equation — like x = 1 or x = 4 — instead of x-values where Marcus's computation is wrong. The task is only about whether Marcus's shown values match $2(x + 3)$, not about which x solves the equation.

The idea underneath. To check someone else's substitution work, recompute each row yourself and compare to what they wrote.

No listed value makes the equation true. The correct response is an empty selection.

1. 1Look at the equation. A solution is a value of x that makes both sides equal.
2. 2$4x$ means 4 times $x$. Dividing both sides by 4 keeps them equal and leaves $x = 28 \div 4 = 7$.
3. 3The true solution is 7, but 7 is not listed. Check the nearest candidates to confirm none work.
4. 4None of the listed values make 4x equal 28. Select nothing.

Common mistake. A common mistake is selecting 28 because it appears in the equation. The solution is the value of x that makes 4x equal 28, which is 7 — but 7 is not among the choices.

The idea underneath. A solution to an equation is the value that makes both sides equal when you substitute it in.