Step-by-Step Guidance for College Algebra: Solving Equations and Interpreting Functions

Q1. Solve the following equation:

$x - \frac{7}{8} = 3x + \frac{1}{10}$

Background

Topic: Solving Linear Equations

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

Key Terms and Formulas:

• Linear Equation: An equation of the form $ax + b = cx + d$.

• Isolate the variable: Get all terms with $x$ on one side and constants on the other.

Step-by-Step Guidance

1. Move all terms involving $x$ to one side and constants to the other. Subtract $3x$ from both sides.

2. Combine like terms to simplify the equation.

3. Add $\frac{7}{8}$ to both sides to move constants together.

4. Combine the fractions on one side. Find a common denominator if needed.

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Q2. Solve the equation:

$-x = 1 - \frac{5(x-5)}{6}$

Background

Topic: Solving Linear Equations with Fractions

This question tests your ability to solve equations with variables inside fractions and requires distributing and combining like terms.

Key Terms and Formulas:

• Distributive Property: $a(b + c) = ab + ac$

• Combining Like Terms: Add or subtract terms with the same variable.

Step-by-Step Guidance

1. Distribute the $5$in the numerator: $5(x-5) = 5x - 25$.

2. Rewrite the equation: $-x = 1 - \frac{5x - 25}{6}$.

3. To eliminate the fraction, multiply both sides by $6$.

4. Expand and simplify both sides after multiplying.

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Q3. Solve the equation:

$\frac{x}{4} - \frac{5}{7} = \frac{1}{3}$

Background

Topic: Solving Linear Equations with Fractions

This question tests your ability to solve equations with fractions by finding a common denominator and isolating the variable.

Key Terms and Formulas:

• Common Denominator: The least common multiple of the denominators.

• Isolate the variable: Get $x$ alone on one side.

Step-by-Step Guidance

1. Add $\frac{5}{7}$ to both sides to move constants together.

2. Combine the right side into a single fraction by finding a common denominator.

3. Multiply both sides by $4$to solve for$x$.

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Q4. Solve the equation:

$4[3(x-13) - 2(2x+2)] = 5[x-6 - 3(3x-4)]$

Background

Topic: Solving Multi-Step Linear Equations

This question tests your ability to use the distributive property, combine like terms, and solve for $x$ in a multi-step equation.

Key Terms and Formulas:

• Distributive Property: $a(b + c) = ab + ac$

• Combine Like Terms: Simplify expressions by adding/subtracting similar terms.

Step-by-Step Guidance

1. Apply the distributive property to expand all brackets on both sides.

2. Simplify each side by combining like terms.

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

4. Solve for $x$ by isolating it.

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Q5. For the function $f(x) = \frac{8}{3} - 8x$, answer the following:

• (a) What is the solution to $f(x) = 0$?

• (b) What is the $x$-intercept?

• (c) What is the zero of $f(x)$?

Background

Topic: Zeros and Intercepts of Linear Functions

This question tests your understanding of how to find the zero of a function, which is the same as the $x$-intercept and the solution to $f(x) = 0$.

Key Terms and Formulas:

• Zero of a function: The value of $x$ where $f(x) = 0$.

• $x$-intercept: The point where the graph crosses the $x$-axis, i.e., $y = 0$.

Step-by-Step Guidance

1. Set $f(x) = 0$ and write the equation: $\frac{8}{3} - 8x = 0$.

2. Add $8x$ to both sides to isolate the constant.

3. Divide both sides by $8$to solve for$x$.

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Q6. Given a table of values for $f(x)$:

X	Y1
6	0
0	-38

• (a) What is the $x$-intercept?

• (b) What is the $y$-intercept?

• (c) What is the solution to $f(x) = 0$?

Background

Topic: Interpreting Function Tables

This question tests your ability to read a table of values and identify intercepts and zeros.

Key Terms and Formulas:

• $x$-intercept: Value of $x$ when $y = 0$.

• $y$-intercept: Value of $y$ when $x = 0$.

• Zero of a function: $x$ such that $f(x) = 0$.

Step-by-Step Guidance

1. Look for the row where $Y1 = 0$ to find the $x$-intercept.

2. Look for the row where $X = 0$ to find the $y$-intercept.

3. The solution to $f(x) = 0$ is the $x$ value where $Y1 = 0$.

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Q7. For the function $f(x)$, given that the zero of the function is $x = 40$:

• (a) What is the $x$-intercept of the graph?

• (b) What is the solution to $f(x) = 0$?

Background

Topic: Zeros and Intercepts of Functions

This question tests your understanding that the zero of a function and the $x$-intercept are the same, and both are solutions to $f(x) = 0$.

Key Terms and Formulas:

• Zero of a function: $x$ such that $f(x) = 0$.

• $x$-intercept: The $x$ value where the graph crosses the $x$-axis.

Step-by-Step Guidance

1. Recall that the zero of the function is the $x$-intercept.

2. The solution to $f(x) = 0$ is the same as the zero.

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Q8. For the function $f(x) = 190 + 20x$:

• (a) What is the zero of $f(x)$?

• (b) What is the $x$-intercept?

• (c) What is the solution to $f(x) = 0$?

Background

Topic: Zeros and Intercepts of Linear Functions

This question tests your ability to find the zero, $x$-intercept, and solution to $f(x) = 0$ for a linear function.

Key Terms and Formulas:

• Zero of a function: $x$ such that $f(x) = 0$.

• $x$-intercept: The $x$ value where the graph crosses the $x$-axis.

Step-by-Step Guidance

1. Set $f(x) = 0$ and write the equation: $190 + 20x = 0$.

2. Subtract $190$from both sides to isolate the$x$ term.

3. Divide both sides by $20$to solve for$x$.

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Q9. Solve the equation using a graphical method:

$16x - 49 = 20 - 7x$

Background

Topic: Solving Linear Equations Graphically

This question tests your ability to solve equations by graphing both sides and finding their intersection.

Key Terms and Formulas:

• Graphical Solution: Plot $y_1 = 16x - 49$ and $y_2 = 20 - 7x$ and find where they intersect.

Step-by-Step Guidance

1. Rewrite the equation as $16x - 49 = 20 - 7x$.

2. Set $y_1 = 16x - 49$ and $y_2 = 20 - 7x$.

3. Graph both equations and find the $x$ value where $y_1 = y_2$.

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Q10. Solve the equation using a graphical method:

$-4 = \frac{3t}{4} + \frac{8t - 1}{3}$

Background

Topic: Solving Equations with Fractions Graphically

This question tests your ability to solve equations with fractions by graphing and finding the intersection point.

Key Terms and Formulas:

• Graphical Solution: Plot $y_1 = -4$ and $y_2 = \frac{3t}{4} + \frac{8t - 1}{3}$ and find their intersection.

Step-by-Step Guidance

1. Set $y_1 = -4$ and $y_2 = \frac{3t}{4} + \frac{8t - 1}{3}$.

2. Graph both equations and find the $t$ value where $y_1 = y_2$.

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Q11. Solve the formula for $t$:

$G = P(1 + rt)$

Background

Topic: Solving Formulas for a Variable

This question tests your ability to rearrange a formula to solve for a specific variable.

Key Terms and Formulas:

• Isolate the variable: Use algebraic operations to get $t$ alone.

Step-by-Step Guidance

1. Divide both sides by $P$ to isolate $(1 + rt)$.

2. Subtract $1$from both sides to isolate$rt$.

3. Divide both sides by $r$ to solve for $t$.

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Q12. Solve for $x$ in the equation $B + H = Mx$:

Background

Topic: Solving Formulas for a Variable

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

Key Terms and Formulas:

• Isolate the variable: Use algebraic operations to get $x$ alone.

Step-by-Step Guidance

1. Subtract $H$ from both sides to isolate $Mx$.

2. Divide both sides by $M$ to solve for $x$.

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Q13. Solve for $y$ and graph the equation:

$9x - 4y = 9$

Background

Topic: Solving Linear Equations for $y$ (Slope-Intercept Form)

This question tests your ability to rearrange a linear equation into $y = mx + b$ form and graph it.

Key Terms and Formulas:

• Slope-Intercept Form: $y = mx + b$

Step-by-Step Guidance

1. Add $4y$ to both sides to move $y$ terms to one side.

2. Subtract $9$from both sides to isolate $4y$.

3. Divide both sides by $-4$ to solve for $y$.

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Q14. The profit from the production and sale of specialty golf hats is given by $P(x) = 20x - 2000$ where $x$ is the number of hats produced and sold.

• (a) Producing and selling how many hats will give a profit of $8000$?

• (b) How many hats must be produced and sold to avoid a loss?

Background

Topic: Solving Linear Equations in Context

This question tests your ability to solve for $x$ in a real-world context using a linear profit function.

Key Terms and Formulas:

• Profit Function: $P(x) = 20x - 2000$

• Set $P(x) = 8000$ for part (a).

• Set $P(x) = 0$ for part (b) to avoid a loss.

Step-by-Step Guidance

1. For part (a), set $P(x) = 8000$ and write the equation: $20x - 2000 = 8000$.

2. Add $2000$to both sides to isolate $20x$.

3. Divide both sides by $20$to solve for$x$.

4. For part (b), set $P(x) = 0$ and solve $20x - 2000 = 0$ using similar steps.

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