Q1. Determine which of the following equations are linear equations in one variable, x:

• A.

• B.

• C.

• D.

Background

Topic: Linear Equations in One Variable

This question tests your ability to identify linear equations in one variable, which is a foundational concept in algebra.

Key Terms and Concepts:

• Linear Equation: An equation where the variable has an exponent of 1 and appears only to the first power.

• One Variable: The equation contains only one variable (e.g., x).

Step-by-Step Guidance

1. Examine each equation to see if it contains only one variable (x).

2. Check if the variable x is raised only to the first power (no exponents other than 1).

3. Eliminate any equation that contains more than one variable or where x is raised to a power other than 1.

4. List all equations that meet both criteria above.

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Q2. Determine whether the following is an expression or an equation:

Background

Topic: Expressions vs. Equations

This question tests your understanding of the difference between an algebraic expression and an equation.

Key Terms:

• Expression: A mathematical phrase that does not include an equality sign (=).

• Equation: A mathematical statement that includes an equality sign (=).

Step-by-Step Guidance

1. Look for the presence of an equality sign (=) in the statement.

2. If there is an equality sign, it is an equation; if not, it is an expression.

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Q3. Determine whether the following is an expression or an equation:

Background

Topic: Expressions vs. Equations

This question checks your ability to distinguish between expressions and equations.

Key Terms:

• Expression: No equality sign (=).

• Equation: Contains an equality sign (=).

Step-by-Step Guidance

1. Check if the statement contains an equality sign (=).

2. If there is no equality sign, it is an expression.

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Q4. Decide whether the following is an expression or an equation:

Background

Topic: Expressions vs. Equations

This question tests your ability to identify expressions.

Key Terms:

• Expression: No equality sign (=).

Step-by-Step Guidance

1. Look for an equality sign (=) in the statement.

2. If there is no equality sign, the statement is an expression.

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Q5. This work contains a common error. Identify the error and give the correct solution:

• Given:

• Step 1: (Distributive property)

• Step 2: (Combine like terms)

• Step 3: (Subtract x, add 9)

Background

Topic: Distributive Property and Combining Like Terms

This question tests your ability to apply the distributive property correctly and combine like terms.

Key Concepts:

• Distributive Property:

• Combining Like Terms: Add or subtract coefficients of the same variable.

Step-by-Step Guidance

1. Carefully apply the distributive property to .

2. Check the sign when distributing the negative.

3. Combine like terms on the left side after distribution.

4. Compare your result to the steps shown to identify where the error occurred.

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Q6. Solve the equation and check the solution. If applicable, tell whether the equation is an identity or a contradiction:

Background

Topic: Solving Linear Equations

This question tests your ability to solve a basic linear equation and classify the solution.

Key Steps:

• Isolate the variable x.

• Check if the equation is always true (identity), never true (contradiction), or has a unique solution.

Step-by-Step Guidance

1. Subtract 6 from both sides to isolate the term with x.

2. Divide both sides by 4 to solve for x.

3. Check your solution by substituting back into the original equation.

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Q7. Solve the equation:

Background

Topic: Solving Linear Equations

This question tests your ability to solve for x in a linear equation.

Key Steps:

• Collect like terms on one side.

• Isolate x.

Step-by-Step Guidance

1. Subtract 12x from both sides to get all x terms on one side.

2. Add 9 to both sides to isolate the variable term.

3. Divide by the coefficient of x to solve for x.

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Q8. Solve:

Background

Topic: Solving Linear Equations

This question tests your ability to combine like terms and solve for x.

Key Steps:

• Combine like terms on each side.

• Isolate x.

Step-by-Step Guidance

1. Combine and on the right side.

2. Subtract (and ) from both sides to get all x terms on one side.

3. Add or subtract constants to isolate x.

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Q9. Solve the equation and check the solution. If applicable, tell whether the equation is an identity or a contradiction:

Background

Topic: Solving Linear Equations

This question tests your ability to combine like terms and solve for w.

Key Steps:

• Combine like terms on both sides.

• Isolate w.

Step-by-Step Guidance

1. Add and on the left side.

2. Simplify constants on both sides.

3. Move all terms with w to one side and constants to the other.

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