Q1. Solve the system of equations using substitution:

Background

Topic: Solving Systems of Linear Equations (Substitution Method)

This question tests your ability to solve a system of two linear equations by substitution, a fundamental skill in algebra.

Key Terms and Formulas:

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

• Substitution method: Solving one equation for a variable and substituting that expression into the other equation.

Step-by-Step Guidance

1. Solve the first equation for (already done): .

2. Set the two expressions for equal to each other: .

3. Combine like terms and solve for . Start by adding to both sides to get all terms on one side.

4. Once you have , substitute it back into either original equation to solve for .

Try solving on your own before revealing the answer!

Q2. Solve the system by any method (graphing, substitution, or elimination):

Background

Topic: Solving Systems of Linear Equations

This question tests your ability to solve a system of equations using your preferred method: graphing, substitution, or elimination.

Key Terms and Formulas:

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

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

• Graphing: Plot both lines and find their intersection.

Step-by-Step Guidance

1. Consider multiplying the second equation by 5 to eliminate denominators: .

2. Rewrite both equations in standard form and align them for elimination.

3. Subtract or add the equations to eliminate one variable.

4. Solve for the remaining variable, then substitute back to find the other variable.

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Q3. A biker bikes into the wind for 60 miles in 6 hours, then returns with the wind in 5 hours. Find the speed in still air and the wind speed.

Background

Topic: Systems of Equations (Distance, Rate, Time Problems)

This problem involves setting up and solving a system of equations based on the relationship , considering the effect of wind on speed.

Key Terms and Formulas:

• Let = speed in still air, = wind speed.

• With wind: ; Against wind: .

• Distance = Rate × Time:

Step-by-Step Guidance

1. Write two equations: (into the wind), (with the wind).

2. Solve each equation for and .

3. Add the two equations to eliminate and solve for .

4. Subtract one equation from the other to solve for .

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Q4. A nursery mixes rye seed ($4.75/lb) and bluegrass seed ($2.50/lb) to make 120 lbs of a blend selling for $3.81/lb. How much of each type was used?

Background

Topic: Mixture Problems (Systems of Linear Equations)

This question tests your ability to set up and solve a system of equations for a mixture problem involving cost and quantity.

Key Terms and Formulas:

• Let = pounds of rye seed, = pounds of bluegrass seed.

• Total weight:

• Total cost:

Step-by-Step Guidance

1. Write the two equations based on total weight and total cost.

2. Solve the first equation for one variable (e.g., ).

3. Substitute into the second equation and solve for .

4. Once you have , substitute back to find .

Try solving on your own before revealing the answer!

Q5. Graph the system of linear inequalities:

Background

Topic: Graphing Systems of Linear Inequalities

This question tests your ability to graph solution regions for systems of inequalities.

Key Terms and Formulas:

• Linear inequality: An inequality involving a linear function.

• Solution region: The set of points that satisfy all inequalities in the system.

Step-by-Step Guidance

1. Graph the boundary line for (the y-axis). Since the inequality is , shade to the right of the y-axis.

2. Graph the line . Since the inequality is , use a solid line and shade below the line.

3. The solution region is where the shaded areas overlap.

Try graphing on your own before revealing the answer!

Q6. Graph the system of linear inequalities:

Background

Topic: Graphing Systems of Linear Inequalities

This question tests your ability to graph and find the solution region for two inequalities.

Key Terms and Formulas:

• Graph each boundary line: and .

• Use dashed or solid lines depending on the inequality.

Step-by-Step Guidance

1. Graph as a dashed line (since the inequality is strict: ), and shade above the line.

2. Graph as a solid line (since the inequality is ), and shade below the line.

3. The solution region is where the shaded areas overlap.

Try graphing on your own before revealing the answer!

Q7. Determine if the expression is a polynomial. If so, write in standard form, state the degree, and classify as monomial, binomial, or trinomial.

Background

Topic: Polynomials (Classification and Standard Form)

This question tests your understanding of what constitutes a polynomial, how to write it in standard form, and how to determine its degree and classification.

Key Terms and Formulas:

• Polynomial: An expression with non-negative integer exponents.

• Standard form: Terms ordered by degree, highest to lowest.

• Degree: The highest sum of exponents in any term.

• Classification: Monomial (1 term), Binomial (2 terms), Trinomial (3 terms).

Step-by-Step Guidance

1. Check if all exponents are non-negative integers (they are).

2. Write the terms in order of decreasing degree: , , $9$.

3. Find the degree by adding exponents in each term and identifying the highest.

4. Count the number of terms to classify the polynomial.

Try classifying on your own before revealing the answer!

Q8. Subtract the polynomials:

Background

Topic: Polynomial Operations (Subtraction)

This question tests your ability to subtract polynomials and combine like terms.

Key Terms and Formulas:

• Distribute the negative sign to each term in the second polynomial.

• Combine like terms (same variable and exponent).

Step-by-Step Guidance

1. Rewrite the subtraction as addition by distributing the negative: .

2. Combine like terms: terms, terms, and constants.

3. Write the result in standard form.

Try simplifying on your own before revealing the answer!