Q: Which of the following best describes a linear equation with exactly one solution?

The equation simplifies to $$ax = b$$ where $$a \neq 0.$$

Question 1

Which of the following is the correct first step to solve the equation $$2(x - 3) = 0$$?

Accepted Answer

Distribute the 2 to both $$x$$ and $$-3.$$

Question 2

What is the solution to the equation $$3(2 - 5x) = 4x + 25$$?

Accepted Answer

$$x = -1$$

Question 3

When solving a linear equation with fractions, such as $$\frac{1}{4}(x+2) - \frac{1}{3}x = \frac{1}{12}$$, what is the best first step?

Accepted Answer

Multiply both sides by the least common denominator.

Question 4

Solve the equation $$\frac{1}{4}(x+2) - \frac{1}{3}x = \frac{1}{12}$$ for $$x.$$

Accepted Answer

$$x = 1$$

Question 5

Which of the following best describes the process of 'combining like terms' in solving linear equations?

Accepted Answer

Adding or subtracting terms with the same variable and exponent.

Question 6

A chemist is mixing two solutions. The equation $$5(x - 2) = 3x + 6$$ models the amount of a chemical in the mixture. What is the value of $$x$$?

Accepted Answer

$$x = 4$$

Question 7

Which step should be performed immediately after distributing constants when solving a linear equation?

Accepted Answer

Combine like terms.

Question 8

A medical researcher sets up the equation $$\frac{1}{2}(x + 6) = x - 1 to $$model a dosage calculation. What is the solution for $$x$$?

Accepted Answer

$$x = 2$$

Question 9

Which of the following is NOT a valid step in solving a linear equation?

Accepted Answer

Multiplying both sides by zero.

Question 10

How many solutions does the equation $$2(x - 1) = 2x - 2$$ have?

Accepted Answer

Infinitely many solutions

Question 11

A historian is analyzing population growth and uses the equation $$7x + 3 = 2x + 18. $$What is the value of $$x$$?

Accepted Answer

$$x = 5$$

Question 12

Which of the following equations has no solution?

Accepted Answer

$$3(x + 2) = 3x + 5$$

Question 13

When categorizing linear equations, which category describes an equation that is true for all values of $$x$$?

Accepted Answer

Infinitely many solutions

Question 14

Solve for $$x$$: $$\frac{2}{3}x - \frac{1}{2} = \frac{1}{6}.$$

Accepted Answer

$$x = 2$$

Question 15

A business calculates profit using the equation $$4(x - 5) = 2x + 6. $$What is the profit value $$x$$?

Accepted Answer

$$x = 8$$

Question 16

Which of the following is the correct order of steps for solving a linear equation with fractions?

Accepted Answer

Multiply by the least common denominator, distribute constants, combine like terms, group $$x$$ and constants, isolate $$x.$$

Question 17

If you check your solution by replacing $$x in $$the original equation and the equation is not true, what does this indicate?

Accepted Answer

You made a calculation error or the equation has no solution.

Question 18

Solve for $$x$$: $$5x - 2(x + 3) = 4.$$

Accepted Answer

$$x = 2$$

Question 19

Which of the following best describes a linear equation with exactly one solution?

Accepted Answer

The equation simplifies to $$ax = b$$ where $$a 
eq 0.$$

Question 20

A real-world scenario: A car rental company charges a flat fee plus a per-mile rate. The equation $$50 + 0.2x = 70$$ models the total cost. How many miles $$x$$ were driven?

Accepted Answer

$$x = 200$$

Solving Linear Equations: Methods and Applications

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