Step-by-Step Guidance for Precalculus/College Algebra Exam Questions

Q1a. Determine algebraically the number of units that gives the break-even for the product.

Background

Topic: Break-even Analysis (Linear Functions)

This question tests your understanding of how to find the break-even point, where revenue equals cost, using linear equations.

Key Terms and Formulas:

• Revenue function: $R(x) = 81.5x$

• Cost function: $C(x) = 58.5x + 2829$

• Break-even point: The value of $x$ where $R(x) = C(x)$

Step-by-Step Guidance

1. Set the revenue and cost functions equal to each other: $81.5x = 58.5x + 2829$

2. Subtract $58.5x$ from both sides to isolate terms with $x$ on one side: $81.5x - 58.5x = 2829$

3. Simplify the left side to combine like terms: $(81.5 - 58.5)x = 2829$

4. Divide both sides by the coefficient of $x$ to solve for $x$.

Try solving on your own before revealing the answer!

Q1b. Determine algebraically the y-coordinate of the break-even point. Explain what the y-coordinate of the break-even means in terms of this application.

Background

Topic: Interpreting Linear Models

This question asks you to find the value of the revenue (or cost) at the break-even quantity and interpret its meaning in context.

Key Terms and Formulas:

• Break-even y-coordinate: $R(x_{BE})$ or $C(x_{BE})$ where $x_{BE}$ is the break-even quantity found in part (a).

Step-by-Step Guidance

1. Take the value of $x$ you found in part (a) (the break-even quantity).

2. Substitute this $x$ value into either $R(x)$ or $C(x)$ to find the corresponding $y$ value: $R(x_{BE}) = 81.5x_{BE}$ or $C(x_{BE}) = 58.5x_{BE} + 2829$

3. Interpret what this $y$ value represents: In this context, it is the total revenue (or cost) at the break-even point, i.e., the amount of money the company makes or spends when it neither profits nor loses.

Try solving on your own before revealing the answer!

Q1c. Graph both functions R(x) and C(x) and label the lines, and the solution point on the graph.

Background

Topic: Graphing Linear Functions and Interpreting Intersections

This question tests your ability to graph linear functions and identify the intersection point, which represents the break-even point.

Key Terms and Formulas:

• Linear function: $y = mx + b$

• Intersection point: The $(x, y)$ where $R(x) = C(x)$

Step-by-Step Guidance

1. Plot $R(x) = 81.5x$ as a straight line passing through the origin (since $b = 0$).

2. Plot $C(x) = 58.5x + 2829$ as a straight line with a y-intercept at $2829$and a slope of $58.5$.

3. Label both lines clearly as "Revenue" and "Cost" on your graph.

4. Mark the intersection point found in part (a) and label it as the break-even point.

Try sketching the graph and labeling the points before checking your answer!

Q2a. Find the model that gives y, smoking of residents, in percent, as a function of x, the number of years after 1985. Round your model to three decimal places.

Background

Topic: Linear Regression/Modeling from Data

This question asks you to find a linear model (best-fit line) that relates the percent of smokers to the number of years after 1985.

Key Terms and Formulas:

• Linear model: $y = mx + b$

• Slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Intercept: Use a data point and the slope to solve for $b$.

Step-by-Step Guidance

1. Convert each year to $x$ values by subtracting 1985 (e.g., 1998 becomes $x = 13$).

2. Choose two data points to calculate the slope $m$ using the formula above.

3. Use one of the points and the slope to solve for the intercept $b$.

4. Write the linear model $y = mx + b$ and round coefficients to three decimal places.

Try setting up the model before checking your answer!

Q2b. Interpret, in words, what the slope of the model in part a. represents.

Background

Topic: Interpreting Slope in Context

This question tests your ability to interpret the meaning of the slope in a real-world context.

Key Terms:

• Slope ($m$): The rate of change of smoking percentage per year after 1985.

Step-by-Step Guidance

1. Recall that the slope represents the change in $y$ for each unit increase in $x$.

2. In this context, $y$ is the percent of smokers, and $x$ is years after 1985.

3. Express in words: The slope tells you how much the smoking percentage changes each year.

Try writing your interpretation before checking the answer!

Q2c. What does the model estimate the percent of US resident smoking to be in 2025? Show your computations.

Background

Topic: Using Linear Models for Prediction

This question asks you to use your model from part (a) to predict the smoking percentage in a future year.

Key Terms and Formulas:

• Model: $y = mx + b$ (from part a)

• For 2025: $x = 2025 - 1985 = 40$

Step-by-Step Guidance

1. Calculate $x$ for 2025 by subtracting 1985 from 2025.

2. Substitute $x = 40$ into your model $y = mx + b$.

3. Carry out the multiplication and addition to find the estimated $y$ value.

Try plugging in the values and computing before checking the answer!

Q3a. Determine algebraically the equation of the profit function for this product.

Background

Topic: Finding Linear Equations from Two Points

This question tests your ability to find the equation of a linear function given two points.

Key Terms and Formulas:

• Profit function: $P(x) = mx + b$

• Given points: $(232, 321)$ and $(264, 545)$

• Slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Step-by-Step Guidance

1. Calculate the slope $m$ using the two given points.

2. Use one of the points and the slope to solve for the intercept $b$.

3. Write the profit function $P(x) = mx + b$.

Try setting up the equation before checking your answer!

Q3b. Interpret the rate of change of the profit function.

Background

Topic: Interpreting Slope in Context

This question asks you to explain what the slope of the profit function means in this business context.

Key Terms:

• Slope ($m$): The change in profit for each additional unit produced and sold.

Step-by-Step Guidance

1. Recall that the slope represents the change in profit per unit increase in $x$.

2. Express in words: For each additional unit produced and sold, the profit increases by $m$ dollars.

Try writing your interpretation before checking the answer!

Q3c. How many units must be produced and sold to avoid a loss? (If need be, round up to the nearest integer)

Background

Topic: Solving Linear Equations for Zero Profit

This question asks you to find the minimum number of units that must be sold for the profit to be at least zero.

Key Terms and Formulas:

• Profit function: $P(x) = mx + b$ (from part a)

• Set $P(x) = 0$ and solve for $x$.

Step-by-Step Guidance

1. Set the profit function equal to zero: $0 = mx + b$

2. Solve for $x$ by isolating $x$ on one side: $x = -\frac{b}{m}$

3. If the result is not an integer, round up to the nearest whole number since you can't sell a fraction of a unit.

Try solving for $x$ before checking the answer!

Q4. Solve the inequality $-\frac{2}{3}(x-3) > \frac{x}{3} + 1$ for $x$.

Background

Topic: Solving Linear Inequalities

This question tests your ability to solve inequalities involving fractions and variables on both sides.

Key Terms and Formulas:

• Distributive property

• Combining like terms

• Solving inequalities

Step-by-Step Guidance

1. Distribute $-\frac{2}{3}$ to both terms inside the parentheses: $-\frac{2}{3}x + 2$

2. Rewrite the inequality: $-\frac{2}{3}x + 2 > \frac{x}{3} + 1$

3. Move all terms involving $x$ to one side and constants to the other.

4. Combine like terms and solve for $x$.

Try working through the steps before checking the answer!

Q5. Solve the system of linear equations by ELIMINATION method.

$3x - 6y = -9$

$9x = 15 - 3y$

Background

Topic: Solving Systems of Linear Equations (Elimination Method)

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

Key Terms and Formulas:

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

• Standard form: $Ax + By = C$

Step-by-Step Guidance

1. Rewrite the second equation in standard form: $9x + 3y = 15$

2. Multiply the first equation by a suitable number if needed so that the coefficients of $y$ or $x$ are opposites.

3. Add or subtract the equations to eliminate one variable.

4. Solve for the remaining variable.

Try eliminating a variable and solving before checking the answer!

Bonus a. Write the equation of the vertical line passing through the point (5, -7).

Background

Topic: Equations of Vertical and Horizontal Lines

This question tests your understanding of the special forms of equations for vertical and horizontal lines.

Key Terms and Formulas:

• Vertical line: $x = a$ where $a$ is the x-coordinate of the point.

Step-by-Step Guidance

1. Recall that a vertical line has an equation of the form $x = a$.

2. Identify the x-coordinate of the given point.

3. Write the equation using this value.

Try writing the equation before checking the answer!

Bonus b. Write the equation of the horizontal line passing through the point (9, 15).

Background

Topic: Equations of Vertical and Horizontal Lines

This question tests your understanding of the special forms of equations for vertical and horizontal lines.

Key Terms and Formulas:

• Horizontal line: $y = b$ where $b$ is the y-coordinate of the point.

Step-by-Step Guidance

1. Recall that a horizontal line has an equation of the form $y = b$.

2. Identify the y-coordinate of the given point.

3. Write the equation using this value.

Try writing the equation before checking the answer!