BackIntermediate Algebra Exam 1 Review – Step-by-Step Guidance
Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
Q1. Distribute to eliminate the parentheses in the expression −3(2x + y − 5).
Background
Topic: Distributive Property
This question tests your ability to apply the distributive property to simplify algebraic expressions by removing parentheses.
Key Terms and Formulas:
Distributive Property: $a(b + c) = ab + ac$
Step-by-Step Guidance
Identify the term outside the parentheses: $-3$.
Multiply $-3$ by each term inside the parentheses: $2x$, $y$, and $-5$.
Write out each multiplication: $-3 \times 2x$, $-3 \times y$, $-3 \times (-5)$.
Combine the results to form a simplified expression.
Try solving on your own before revealing the answer!
Final Answer: $-6x - 3y + 15$
Each term inside the parentheses is multiplied by $-3$, resulting in $-6x$, $-3y$, and $+15$.
Q2. Add like terms in the expression 3x − 2y − x + 5 − 7 + 4y + 2x − 2.
Background
Topic: Combining Like Terms
This question tests your ability to identify and combine like terms in an algebraic expression.
Key Terms and Formulas:
Like Terms: Terms that have the same variable raised to the same power.
Step-by-Step Guidance
Group all $x$ terms together: $3x$, $-x$, $2x$.
Group all $y$ terms together: $-2y$, $4y$.
Group all constant terms: $5$, $-7$, $-2$.
Add the coefficients for each group to combine like terms.
Try solving on your own before revealing the answer!
Final Answer: $4x + 2y - 4$
All like terms are combined: $3x - x + 2x = 4x$, $-2y + 4y = 2y$, $5 - 7 - 2 = -4$.
Q3. Solve for $x$ from the equation $3x + 2 = 7$.
Background
Topic: Solving Linear Equations
This question tests your ability to isolate the variable $x$ in a simple linear equation.
Key Terms and Formulas:
Linear Equation: An equation of the form $ax + b = c$.
Step-by-Step Guidance
Subtract $2$ from both sides to isolate the term with $x$.
Divide both sides by $3$ to solve for $x$.
Try solving on your own before revealing the answer!
Final Answer: $x = rac{5}{3}$
Subtracting $2$ gives $3x = 5$, then dividing by $3$ yields $x = \frac{5}{3}$.
Q4. Substitute $x = 3$ into the expression $4x - 11$.
Background
Topic: Evaluating Expressions
This question tests your ability to substitute a value for a variable and simplify the expression.
Key Terms and Formulas:
Substitution: Replacing a variable with a given value.
Step-by-Step Guidance
Replace $x$ with $3$ in the expression: $4x - 11$ becomes $4 \times 3 - 11$.
Multiply $4$ by $3$.
Subtract $11$ from the result of the multiplication.
Try solving on your own before revealing the answer!
Final Answer: $1$
Substituting $x = 3$ gives $4 \times 3 - 11 = 12 - 11 = 1$.
Q5. What is the least common denominator of $\frac{1}{6}$, $\frac{1}{8}$, $\frac{1}{12}$?
Background
Topic: Least Common Denominator (LCD)
This question tests your ability to find the least common denominator for a set of fractions.
Key Terms and Formulas:
Least Common Denominator (LCD): The smallest number that is a multiple of all denominators.
Step-by-Step Guidance
List the denominators: $6$, $8$, $12$.
Find the least common multiple (LCM) of these numbers.
Check which is the smallest number divisible by $6$, $8$, and $12$.
Try solving on your own before revealing the answer!
Final Answer: $24$
The LCM of $6$, $8$, and $12$ is $24$.
Q6. Simplify the expression $\frac{2x(x + 2)(2x + 1)}{2x + 1}$.
Background
Topic: Simplifying Rational Expressions
This question tests your ability to simplify rational expressions by canceling common factors.
Key Terms and Formulas:
Rational Expression: A fraction with polynomials in the numerator and denominator.
Canceling: If a factor appears in both numerator and denominator, it can be divided out.
Step-by-Step Guidance
Factor both numerator and denominator if possible.
Identify common factors in the numerator and denominator.
Cancel the common factor $(2x + 1)$ from both numerator and denominator.
Write the simplified expression.
Try solving on your own before revealing the answer!
Final Answer: $2x(x + 2)$
The $(2x + 1)$ cancels, leaving $2x(x + 2)$.
Q7. Solve for $x$ from the equation $\frac{2}{x} = \frac{5}{2}$.
Background
Topic: Solving Rational Equations
This question tests your ability to solve equations involving fractions with variables in the denominator.
Key Terms and Formulas:
Cross Multiplication: For $\frac{a}{b} = \frac{c}{d}$, $ad = bc$.
Step-by-Step Guidance
Set up the equation: $\frac{2}{x} = \frac{5}{2}$.
Cross-multiply: $2 \times 2 = 5 \times x$.
Solve for $x$ by dividing both sides by $5$.
Try solving on your own before revealing the answer!
Final Answer: $x = \frac{4}{5}$
Cross-multiplying gives $4 = 5x$, so $x = \frac{4}{5}$.
Q8. What is the least common denominator of $\frac{1}{2x}$, $\frac{x-1}{x+2}$, $\frac{x+1}{4}$?
Background
Topic: Least Common Denominator (LCD) with Variables
This question tests your ability to find the LCD for rational expressions with variable denominators.
Key Terms and Formulas:
LCD: The smallest expression that is a multiple of all denominators.
Step-by-Step Guidance
List all denominators: $2x$, $x+2$, $4$.
Factor each denominator if possible.
Find the product that includes each unique factor the greatest number of times it appears in any denominator.
Multiply these factors together to get the LCD.
Try solving on your own before revealing the answer!
Final Answer: $4x(x + 2)$
The LCD is the product of $4$, $x$, and $(x + 2)$.
Q9. Expand and simplify the expression $(x + 2)(x - 5)$.
Background
Topic: Multiplying Binomials (FOIL Method)
This question tests your ability to expand and simplify the product of two binomials.
Key Terms and Formulas:
FOIL Method: First, Outer, Inner, Last terms multiplication.
Step-by-Step Guidance
Multiply the first terms: $x \times x$.
Multiply the outer terms: $x \times (-5)$.
Multiply the inner terms: $2 \times x$.
Multiply the last terms: $2 \times (-5)$.
Add all the products together and combine like terms.
Try solving on your own before revealing the answer!
Final Answer: $x^2 - 3x - 10$
Expanding gives $x^2 - 5x + 2x - 10 = x^2 - 3x - 10$.
Q10. Solve the equation $3x^2 + 2x - 1 = 3x^2 + 4x$.
Background
Topic: Solving Quadratic Equations
This question tests your ability to solve quadratic equations by simplifying and isolating $x$.
Key Terms and Formulas:
Quadratic Equation: $ax^2 + bx + c = 0$
Step-by-Step Guidance
Subtract $3x^2$ from both sides to eliminate the $x^2$ terms.
Simplify the equation to isolate terms with $x$.
Solve the resulting linear equation for $x$.
Try solving on your own before revealing the answer!
Final Answer: $x = -1$
After simplifying, you get $2x - 1 = 4x$, so $x = -1$.
Q11. Find the $x$ and $y$ intercepts of the line $2x + 3y = 12$.
Background
Topic: Intercepts of a Line
This question tests your ability to find where a line crosses the $x$-axis and $y$-axis.
Key Terms and Formulas:
$x$-intercept: Set $y = 0$ and solve for $x$.
$y$-intercept: Set $x = 0$ and solve for $y$.
Step-by-Step Guidance
For the $x$-intercept, set $y = 0$ in $2x + 3y = 12$ and solve for $x$.
For the $y$-intercept, set $x = 0$ and solve for $y$.
Try solving on your own before revealing the answer!
Final Answer: $x$-intercept: $(6, 0)$; $y$-intercept: $(0, 4)$
Setting $y = 0$ gives $x = 6$, and $x = 0$ gives $y = 4$.
Q12. Write the line $y - 3 = 5(x + 1)$ in slope-intercept form.
Background
Topic: Slope-Intercept Form
This question tests your ability to rewrite an equation in the form $y = mx + b$.
Key Terms and Formulas:
Slope-Intercept Form: $y = mx + b$
Step-by-Step Guidance
Expand the right side: $5(x + 1)$.
Add $3$ to both sides to solve for $y$.
Simplify to get $y$ by itself.
Try solving on your own before revealing the answer!
Final Answer: $y = 5x + 8$
Expanding and simplifying gives $y = 5x + 8$.
Q13. Find three points on the line $y = 2x - 3$.
Background
Topic: Graphing Linear Equations
This question tests your ability to find points that satisfy a linear equation.
Key Terms and Formulas:
Choose values for $x$, substitute into the equation to find $y$.
Step-by-Step Guidance
Choose three values for $x$ (e.g., $x = 0$, $x = 1$, $x = 2$).
For each $x$, substitute into $y = 2x - 3$ to find the corresponding $y$.
Write each $(x, y)$ pair as a point.
Try solving on your own before revealing the answer!
Final Answer: $(0, -3)$, $(1, -1)$, $(2, 1)$
Substituting $x = 0, 1, 2$ gives the points above.
Q14. Write an equation for the line through $(2, -1)$ and $(2, 5)$.
Background
Topic: Vertical and Horizontal Lines
This question tests your ability to recognize and write the equation of a vertical line.
Key Terms and Formulas:
Vertical Line: $x = a$ for some constant $a$.
Step-by-Step Guidance
Notice both points have the same $x$-coordinate ($x = 2$).
The equation is $x = 2$.
Try solving on your own before revealing the answer!
Final Answer: $x = 2$
This is a vertical line through $x = 2$.
Q15. Write an equation for the line through $(2, -1)$ and $(5, -1)$.
Background
Topic: Vertical and Horizontal Lines
This question tests your ability to recognize and write the equation of a horizontal line.
Key Terms and Formulas:
Horizontal Line: $y = b$ for some constant $b$.
Step-by-Step Guidance
Notice both points have the same $y$-coordinate ($y = -1$).
The equation is $y = -1$.
Try solving on your own before revealing the answer!
Final Answer: $y = -1$
This is a horizontal line through $y = -1$.
Q16. Determine if the system $\begin{cases}2x - 3y = 7 \\ 6x - 9y = 21\end{cases}$ is dependent, independent, or inconsistent.
Background
Topic: Systems of Linear Equations
This question tests your ability to classify a system as dependent, independent, or inconsistent by comparing the equations.
Key Terms and Formulas:
Dependent: Infinite solutions (same line).
Independent: One solution (intersecting lines).
Inconsistent: No solution (parallel lines).
Step-by-Step Guidance
Compare the two equations to see if one is a multiple of the other.
Multiply the first equation by $3$ and compare to the second equation.
Check if both sides match exactly.
Try solving on your own before revealing the answer!
Final Answer: Dependent
The second equation is exactly $3$ times the first, so the system is dependent (infinite solutions).
Q17. Determine if the system $\begin{cases}2x - 3y = 7 \\ 6x - 9y = 10\end{cases}$ is dependent, independent, or inconsistent.
Background
Topic: Systems of Linear Equations
This question tests your ability to classify a system by comparing the ratios of coefficients and constants.
Key Terms and Formulas:
Compare ratios: If the ratios of coefficients are equal but the constants are not, the system is inconsistent.
Step-by-Step Guidance
Multiply the first equation by $3$ to compare with the second equation.
Check if the left sides are proportional but the right sides are not.
If so, the system is inconsistent.
Try solving on your own before revealing the answer!
Final Answer: Inconsistent
The left sides are proportional, but the constants are not, so there is no solution.
Q18. Determine if the system $\begin{cases}2x - 3y = 7 \\ 6x + 9y = 21\end{cases}$ is dependent, independent, or inconsistent.
Background
Topic: Systems of Linear Equations
This question tests your ability to compare the ratios of coefficients and constants to classify the system.
Key Terms and Formulas:
Independent: If the ratios of coefficients are not equal, the system is independent (one solution).
Step-by-Step Guidance
Compare the ratios of $x$ and $y$ coefficients in both equations.
If the ratios are not equal, the lines intersect at one point (independent).
Try solving on your own before revealing the answer!
Final Answer: Independent
The ratios are not equal, so the system has one solution.
Q19. Substitute $y = 3$ into the equation $2x - 5y = 11$.
Background
Topic: Substitution in Linear Equations
This question tests your ability to substitute a value for a variable and solve for the other variable.
Key Terms and Formulas:
Substitute $y = 3$ into $2x - 5y = 11$.
Step-by-Step Guidance
Replace $y$ with $3$ in the equation: $2x - 5 \times 3 = 11$.
Multiply $-5$ by $3$.
Add $15$ to both sides to solve for $2x$.
Divide both sides by $2$ to solve for $x$.
Try solving on your own before revealing the answer!
Final Answer: $x = 13$
Substituting and solving gives $x = 13$.
Q20. Solve the system of equations $\begin{cases}x - 3y = -5 \\ y = 4\end{cases}$.
Background
Topic: Solving Systems by Substitution
This question tests your ability to solve a system of equations using substitution.
Key Terms and Formulas:
Substitute the value of $y$ into the first equation to solve for $x$.
Step-by-Step Guidance
Substitute $y = 4$ into $x - 3y = -5$.
Multiply $-3$ by $4$.
Add $12$ to both sides to solve for $x$.
Try solving on your own before revealing the answer!
Final Answer: $x = 7$, $y = 4$
Substituting $y = 4$ gives $x = 7$.
Q21. Solve the 1st order equation $3(x - 1) + 2(3x - 5) = 5(2x + 3)$.
Background
Topic: Solving Linear Equations
This question tests your ability to use the distributive property and combine like terms to solve for $x$.
Key Terms and Formulas:
Distributive Property: $a(b + c) = ab + ac$
Step-by-Step Guidance
Distribute $3$ to $(x - 1)$ and $2$ to $(3x - 5)$.
Distribute $5$ to $(2x + 3)$.
Combine like terms on both sides.
Isolate $x$ on one side of the equation.
Try solving on your own before revealing the answer!
Final Answer: $x = 4$
After simplifying and solving, $x = 4$.
Q22. Solve the rational equation $\frac{1}{3x} + \frac{2}{x + 1} = \frac{1}{6x}$.
Background
Topic: Solving Rational Equations
This question tests your ability to find a common denominator and solve for $x$.
Key Terms and Formulas:
LCD: Least common denominator for all fractions.
Step-by-Step Guidance
Identify the LCD for $3x$, $x + 1$, and $6x$.
Multiply both sides by the LCD to clear denominators.
Solve the resulting equation for $x$.
Try solving on your own before revealing the answer!
Final Answer: $x = 1$
After clearing denominators and solving, $x = 1$.
Q23. Write the line $5x - 3y = 30$ in slope-intercept form.
Background
Topic: Slope-Intercept Form
This question tests your ability to solve for $y$ and rewrite the equation as $y = mx + b$.
Key Terms and Formulas:
Slope-Intercept Form: $y = mx + b$
Step-by-Step Guidance
Subtract $5x$ from both sides to isolate terms with $y$.
Divide both sides by $-3$ to solve for $y$.
Simplify the equation to $y = mx + b$ form.
Try solving on your own before revealing the answer!
Final Answer: $y = \frac{5}{3}x - 10$
Rewriting gives $y = \frac{5}{3}x - 10$.
Q24. Solve the system of equations $\begin{cases}2x - 3y = 3 \\ 2x + 3y = 9\end{cases}$.
Background
Topic: Solving Systems by Addition/Elimination
This question tests your ability to solve a system using the elimination method.
Key Terms and Formulas:
Add or subtract equations to eliminate one variable.
Step-by-Step Guidance
Add the two equations to eliminate $y$.
Solve for $x$.
Substitute $x$ back into one equation to solve for $y$.
Try solving on your own before revealing the answer!
Final Answer: $x = 3$, $y = 1$
Adding the equations gives $4x = 12$, so $x = 3$, then $y = 1$.
Q25. Solve the 1st order equation $\frac{x}{2} - \frac{2}{7} = 7x + \frac{4}{2}$.
Background
Topic: Solving Linear Equations with Fractions
This question tests your ability to clear fractions and solve for $x$.
Key Terms and Formulas:
Multiply both sides by the LCD to clear denominators.
Step-by-Step Guidance
Find the LCD for all denominators ($2$ and $7$).
Multiply both sides by the LCD to eliminate fractions.
Solve the resulting equation for $x$.
Try solving on your own before revealing the answer!
Final Answer: $x = -1$
After clearing denominators and solving, $x = -1$.
Q26. Solve the rational equation $\frac{1}{x + 2} + \frac{2}{x - 1} = \frac{3}{x}$.
Background
Topic: Solving Rational Equations
This question tests your ability to find the LCD and solve for $x$.
Key Terms and Formulas:
LCD: Product of all unique denominators.
Step-by-Step Guidance
Identify the LCD: $(x + 2)(x - 1)x$.
Multiply both sides by the LCD to clear denominators.
Solve the resulting equation for $x$.
Try solving on your own before revealing the answer!
Final Answer: $x = 2$
After clearing denominators and solving, $x = 2$.
Q27. Write the point-slope form of the line through the points $(-2, 3)$ and $(3, -1)$. Find the $x$ and $y$ intercepts of the line.
Background
Topic: Point-Slope Form and Intercepts
This question tests your ability to find the equation of a line given two points and to find its intercepts.
Key Terms and Formulas:
Slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$
Point-Slope Form: $y - y_1 = m(x - x_1)$
Intercepts: Set $x = 0$ for $y$-intercept, $y = 0$ for $x$-intercept.
Step-by-Step Guidance
Calculate the slope $m$ using the two points.
Write the point-slope form using one of the points and the slope.
To find the $y$-intercept, set $x = 0$ and solve for $y$.
To find the $x$-intercept, set $y = 0$ and solve for $x$.
Try solving on your own before revealing the answer!
Final Answer:
Point-slope form: $y + 1 = -\frac{4}{5}(x - 3)$ or $y - 3 = -\frac{4}{5}(x + 2)$
$y$-intercept: $(0, \frac{1}{2})$; $x$-intercept: $(\frac{3}{2}, 0)$
Q28. Solve the system of equations $\begin{cases}3x + 4y = 5 \\ 2x + 3y = 11\end{cases}$.
Background
Topic: Solving Systems by Elimination
This question tests your ability to solve a system of equations using elimination or substitution.
Key Terms and Formulas:
Multiply equations to align coefficients for elimination.
Solve for one variable, then substitute back to find the other.
Step-by-Step Guidance
Multiply one or both equations to align coefficients for elimination.
Subtract or add equations to eliminate one variable.
Solve for the remaining variable.
Substitute back to find the other variable.
Try solving on your own before revealing the answer!
Final Answer: $x = -29$, $y = 22$
After elimination and substitution, $x = -29$, $y = 22$.
