Q1. Solve the system of equations:

\[ \begin{cases} 2x + 5y = 2 \\ 3x - 4y = 20 \end{cases} \]

Background

Topic: Systems of Linear Equations

This question tests your ability to solve a system of two linear equations using either the substitution or elimination method.

Key Terms and Formulas

• System of equations: Two or more equations with the same variables.

• Elimination method: Add or subtract equations to eliminate one variable.

• Substitution method: Solve one equation for one variable and substitute into the other.

Step-by-Step Guidance

1. Choose a variable to eliminate. For example, multiply the first equation by 4 and the second by 5 to align the coefficients of .

2. Write the new system:

3. Simplify both equations and add or subtract to eliminate .

4. Solve for in the resulting equation.

Try solving on your own before revealing the answer!

Q2. Solve the system:

\[ \begin{cases} x + y + z = 6 \\ 3x + 4y - z = 5 \\ 2x - y + 3z = 8 \end{cases} \]

Background

Topic: Systems of Three Linear Equations

This question tests your ability to solve a system of three equations with three variables, often using substitution or elimination.

Key Terms and Formulas

• System of three equations: Three equations with three variables (, , ).

• Elimination/Substitution: Methods to reduce the system to two equations with two variables, then solve.

Step-by-Step Guidance

1. Choose two equations and eliminate one variable (e.g., ) by adding or subtracting equations.

2. Repeat with a different pair to get a second equation in two variables.

3. Solve the resulting two-variable system for and .

4. Substitute back to find .

Try solving on your own before revealing the answer!

Q3. Find the distance between the points (-2, 3) and (3, 9).

Background

Topic: Distance Formula

This question tests your ability to use the distance formula to find the length between two points in the coordinate plane.

Key Terms and Formulas

• Distance formula:

Step-by-Step Guidance

1. Label the points: and .

2. Identify , , , .

3. Plug these values into the distance formula.

4. Simplify inside the square root before calculating the final value.

Try solving on your own before revealing the answer!

Q4. Find the midpoint between (2, 6) and (-1, 4).

Background

Topic: Midpoint Formula

This question tests your ability to find the midpoint of a segment connecting two points in the coordinate plane.

Key Terms and Formulas

• Midpoint formula:

Step-by-Step Guidance

1. Label the points: and .

2. Add the -coordinates and divide by 2.

3. Add the -coordinates and divide by 2.

Try solving on your own before revealing the answer!

Q5. Given , find:

• a)

• b)

Background

Topic: Evaluating Functions

This question tests your ability to substitute values into a quadratic function and simplify.

Key Terms and Formulas

• Function evaluation: Substitute the given value for in .

Step-by-Step Guidance

1. For , substitute into : .

2. Simplify the expression step by step.

3. Repeat for : .

Try solving on your own before revealing the answer!

Q6. Given , find:

• a)

• b)

Background

Topic: Evaluating Linear Functions

This question tests your ability to substitute values into a linear function and simplify.

Key Terms and Formulas

• Function evaluation: Substitute the given value for in .

Step-by-Step Guidance

1. For , substitute into : .

2. Simplify the expression step by step.

3. Repeat for : .

Try solving on your own before revealing the answer!

Q7. Given , find:

• a)

• b)

• c)

Background

Topic: Piecewise Functions

This question tests your ability to evaluate a piecewise function by determining which case applies for each input value.

Key Terms and Formulas

• Piecewise function: A function defined by different expressions depending on the input value.

Step-by-Step Guidance

1. For each input, determine if or to select the correct formula.

2. Substitute the value into the appropriate expression and simplify.

Try solving on your own before revealing the answer!

Q8. For the given graph, find the domain, range, x-intercepts, y-intercepts, and intervals where the function is increasing or decreasing.

Background

Topic: Graph Analysis

This question tests your ability to interpret a graph and extract key features such as domain, range, intercepts, and intervals of increase/decrease.

Key Terms and Concepts

• Domain: All possible -values.

• Range: All possible -values.

• x-intercepts: Points where the graph crosses the -axis.

• y-intercepts: Points where the graph crosses the -axis.

• Increasing/Decreasing: Intervals where the function rises or falls as increases.

Step-by-Step Guidance

1. Examine the graph to determine the leftmost and rightmost -values (domain).

2. Identify the lowest and highest -values (range).

3. Find where the graph crosses the -axis and -axis.

4. Look for intervals where the graph moves upward (increasing) or downward (decreasing).

Try solving on your own before revealing the answer!