Quiz: Mastering Linear Equations and Word Problems — 10 domande

1. Why is it useful to combine like terms before solving an equation such as ax+b=cx+d?

A. It removes the need to use inverse operations

B. It guarantees the equation has exactly one solution

C. It turns the equation into a simpler form with fewer terms to move

D. It changes every variable term into a constant

Answer: (C)

Spiegazione

Combining like terms simplifies the equation and makes it easier to move variable terms and constants to opposite sides. It does not automatically determine the number of solutions.

2. If a rectangle’s length is 3 cm more than twice its width, which equation models that relationship?

A. L=2(W+3)

B. L=3W+2

C. L=2W+3

D. W=2L+3

Answer: (C)

Spiegazione

“3 cm more than twice its width” translates to twice the width plus 3, so L=2W+3. The other options change the meaning of the statement.

3. Which expression best represents a cost model with a starting fee plus a per-unit rate?

A. starting fee × rate + number of units

B. rate + starting fee × number of units

C. starting fee - rate × number of units

D. starting fee + rate × number of units

Answer: (D)

Spiegazione

A cost model is written as a fixed starting fee plus a rate multiplied by the number of units. This structure appears in taxis, phone plans, and gym memberships.

4. When x appears on both sides of an equation, what is the standard next step before solving the final one-step equation?

A. Move every term to the side with the larger coefficient

B. Replace x with zero to simplify both sides

C. Gather all x-terms on one side and all constants on the other

D. Multiply each side by x

Answer: (C)

Spiegazione

The key strategy is to collect variable terms on one side and constants on the other so the equation can be simplified. This creates a one-step equation that is easier to solve.

5. In a multi-step linear equation, what should be done before combining terms when parentheses are present on both sides?

A. Divide by the coefficient of x

B. Distribute across every set of parentheses

C. Substitute a trial value for x

D. Move all constants to the left side

Answer: (B)

Spiegazione

Distribution must happen first so all terms are written plainly before combining like terms. This is especially important in expressions with multiple parentheses.

6. What does it mean to isolate the variable in a linear equation?

A. To expand every expression before collecting terms

B. To multiply both sides by the variable

C. To move every constant to the left side and leave the variable everywhere

D. To rewrite the equation so the variable is alone on one side

Answer: (D)

Spiegazione

Isolating the variable means rewriting the equation so the variable stands by itself on one side. This is the final goal after combining like terms and using inverse operations.

7. Why should expressions such as (x-10) or (2x+5) be distributed before combining terms in a fractional linear equation?

A. Because the fractions must be removed before any subtraction can happen

B. Because the variable must be isolated before the equation is rewritten

C. Because the distributive property is needed to expand the parentheses correctly

D. Because like terms cannot be combined until after the answer is found

Answer: (C)

Spiegazione

When parentheses contain fractional expressions, distributing first turns the equation into a form where terms can be combined correctly. Combining too early can hide terms and lead to mistakes.

8. Which equation correctly represents the perimeter of a rectangle with length L and width W?

A. L×W

B. L-W

C. 2(L+W)

D. L+W

Answer: (C)

Spiegazione

The perimeter of a rectangle is the total distance around it, which is twice the sum of its length and width. Area would be L×W, not perimeter.

9. How should three consecutive integers be represented in an integer equation?

A. n, n+1, n+2

B. n-1, n, n+1

C. n, 2n, 3n

D. n-2, n, n+2

Answer: (A)

Spiegazione

Consecutive integers increase by 1 each time, so they are represented as n, n+1, and n+2. The other options describe different number patterns.

10. What is the first best step when solving a linear equation that contains fractions on both sides?

A. Add the denominators to both sides

B. Move every x-term to the right side first

C. Multiply both sides by the least common denominator

D. Replace each fraction with a decimal estimate

Answer: (C)

Spiegazione

Multiplying by the least common denominator clears the fractions efficiently so the equation can be solved more easily. Moving x-terms first may still leave fractions in the equation.