Solving linear equations: Algebra 1 Regents Practice

Solve $5(x - 2) \le 3x + 20$ algebraically. Show your work.

Solve an inequality just like an equation, only flip the inequality sign if you multiply or divide by a negative. Here you divide by +2, so it stays ≤.

Solve for $x$: $2x + 7 = 19$.

• (A) $x = 13$
• (B) $x = 26$
• (C) $x = 12$
• (D) $x = 6$ ✓

Undo operations in reverse: subtract the constant, then divide by the coefficient.

Solve for $x$: $4x + 8 = 32$.

• (A) $x = 24$
• (B) $x = 8$
• (C) $x = 10$
• (D) $x = 6$ ✓

A two-step linear equation is solved by first undoing addition or subtraction, then undoing multiplication or division.

Solve for $x$: $7x - 5 = 3x + 15$.

• (A) $x = 5$ ✓
• (B) $x = 2.5$
• (C) $x = 20$
• (D) $x = 2$

When variables appear on both sides, collect the variable terms on one side and the constants on the other, then divide.

What is the value of $x$ in the equation $3(x - 4) = 18$?

• (A) $x = 2$
• (B) $x = 6$
• (C) $x = 10$ ✓
• (D) $x = 30$

Dividing both sides by $3$ gives $x - 4 = 6$. Adding $4$ to both sides isolates $x$, so $x = 10$.

What is the value of $x$ in the equation $3(x - 4) = 2x + 6$?

• (A) $10$
• (B) $2$
• (C) $-6$
• (D) $18$ ✓

Distribute the 3 across the parentheses to get $3x - 12 = 2x + 6$, then collect like terms: $x = 18$. The answer $10$ results from forgetting to multiply the $-4$ by 3, and $-6$ results from a sign error when moving the $-12$ across the equals sign.

What is the solution to the equation $3(x - 4) = 2x + 6$?

• (A) $x = 2$
• (B) $x = 10$
• (C) $x = -6$
• (D) $x = 18$ ✓

Distributing correctly gives $3x - 12$ on the left side. Subtracting $2x$ and then adding $12$ isolates $x$, giving $x = 18$. Students who distribute only to the first term, writing $3x - 4$, get $x = 10$, and sign errors lead to the other distractors.

Solve the equation $3(x-4)+5=2x+7$ algebraically for $x$.

Distributing and combining like terms simplifies the left side to $3x-7$. Then using the properties of equality to collect the variable terms on one side and the constants on the other gives $x=14$, which checks: $3(14-4)+5=35$ and $2(14)+7=35$.

What is the value of $x$ in the equation $3(x - 4) + 5 = 2x + 7$?

• (A) $6$
• (B) $14$ ✓
• (C) $2$
• (D) $24$

Distributing correctly gives $3x - 12$, not $3x - 4$. After combining constants, the left side is $3x - 7$, and isolating $x$ yields $x = 14$. Checking, $3(10) + 5 = 35$ and $2(14) + 7 = 35$.

What is the value of $x$ in the equation $4(x - 3) = 2x + 10$?

• (A) $-1$
• (B) $6.5$
• (C) $5$
• (D) $11$ ✓

Distributing the 4 gives $4x - 12 = 2x + 10$. Collecting the variable terms gives $2x = 22$, so $x = 11$. The distractor $6.5$ comes from distributing only to the $x$ term ($4x - 3$), and $-1$ comes from a sign error when moving the constant.

Solve the equation $\frac{2}{3}(x + 9) = 5x - 7$ algebraically for $x$.

Clearing the fraction first by multiplying both sides by 3 keeps the algebra clean. Distributing and collecting variable terms on one side gives $13x = 39$, so $x = 3$, which checks in the original equation since both sides equal 8.

Solve the equation $5(x - 2) - 3x = 3x + 8$ algebraically for $x$. Show each step of your solution.

Applying the distributive property and combining like terms reduces the equation to a simple two-step linear equation. Collecting the x terms on one side gives $x = -18$, and substituting back into the original equation makes both sides equal $-46$, confirming the solution.

What is the solution to $\dfrac{4(x - 5)}{3} + 2 = 14$ ?

• (A) $x = 4$
• (B) $x = 14$ ✓
• (C) $x = 6$
• (D) $x = 15$

Clear the fraction and the added constant first, then peel back the remaining operations one at a time.

Chloe begins solving $x^{2}+5x=3x+3$ and writes $x^{2}+2x-3=0$ as her first step. Which property justifies this step?

• (A) the zero product property
• (B) the distributive property
• (C) the subtraction property of equality ✓
• (D) the commutative property

Moving terms to one side by subtracting the same quantity from both sides is the subtraction property of equality.

The solution to $\dfrac{2(3x-1)}{3}=x+2$ is

• (A) $\tfrac{1}{3}$
• (B) $\tfrac{8}{3}$ ✓
• (C) $\tfrac{4}{3}$
• (D) $\tfrac{2}{3}$

Clear the denominator, distribute, gather x-terms on one side, and divide.

Solve for $x$: $3(2x - 4) = 2(x + 6)$.

• (A) $x = 4$
• (B) $x = 3$
• (C) $x = 6$ ✓
• (D) $x = 0$

Distribute completely on both sides before collecting like terms, making sure each term inside the parentheses is multiplied.

Solve for $x$: $\dfrac{x + 4}{2} = 6$.

• (A) $x = 8$ ✓
• (B) $x = 16$
• (C) $x = 2$
• (D) $x = 4$

When an entire expression sits over a denominator, clear the fraction by multiplying both sides by that denominator before doing anything else.

What is the value of $x$ in the equation $3(x - 4) = 2x + 5$?

• (A) $17$ ✓
• (B) $9$
• (C) $5$
• (D) $-7$

After distributing the 3 you must add 12 (not subtract it) when isolating $x$, giving $x = 17$. Forgetting to distribute the 3 to the $-4$ is the most common mistake.

Solve the equation $5(x-3)-2x=4x-27$ algebraically for $x$. Justify each step of your solution.

The key idea is applying properties of equality after distributing and combining like terms. Each operation keeps the equation balanced, and substituting $x=12$ back in gives $5(9)-24=45-24=21$ on the left and $48-27=21$ on the right, confirming the solution.

When solved for $x$ in terms of $a$, the solution to $3x - 7 = ax + 5$ is

• (A) $x = \dfrac{3a}{12}$
• (B) $x = \dfrac{12}{3a}$
• (C) $x = \dfrac{3 - a}{12}$
• (D) $x = \dfrac{12}{3 - a}$ ✓

Solving 'in terms of a' is ordinary equation-solving with a letter in place of a number: collect the variable, factor it out, then divide.

The formula $P = 2l + 2w$ gives the perimeter of a rectangle. Solve this equation for $w$.

• (A) $w = P - 2l$
• (B) $w = \dfrac{2l - P}{2}$
• (C) $w = \dfrac{P}{2} - 2l$
• (D) $w = \dfrac{P - 2l}{2}$ ✓

Solving a literal equation uses the same inverse operations as a numeric one: isolate the target variable while treating the other letters as constants.

Solve for $x$: $\dfrac{1}{3}(9x - 6) = 2x + 1$.

• (A) $x = -1$
• (B) $x = 3$ ✓
• (C) $x = 1$
• (D) $x = 7$

Distribute the fraction to every term inside the parentheses first, then collect like terms across the equal sign.

What is the value of $x$ in the equation $\dfrac{3x - 2}{4} = \dfrac{x + 5}{2}$?

• (A) $11$
• (B) $8$
• (C) $24$
• (D) $12$ ✓

Clear the denominators by cross multiplying, then distribute and collect like terms to get $2x = 24$, so $x = 12$. A check confirms it: $\frac{3(12)-2}{4} = \frac{34}{4} = 8.5$ and $\frac{12+5}{2} = \frac{17}{2} = 8.5$. Failing to divide by $2$ at the end gives $24$, a distribution slip gives $11$, and a sign error when moving the constant gives $8$.