Q1. Determine if each ordered triple is a solution of the given system.

Background

Topic: Systems of Linear Equations in Three Variables

This question tests your ability to substitute ordered triples into a system of three equations and check if they satisfy all equations.

Key Terms and Formulas

• Ordered triple: A set of three numbers (x, y, z) representing a solution in three variables.

• System of equations: Multiple equations that are solved together because they share variables.

Step-by-Step Guidance

1. Write out the system of equations clearly:

2. For the first ordered triple (3, -1, 4):

   • Substitute , , into each equation one at a time.

3. Check if the values satisfy all three equations. If they do, the triple is a solution; if not, it is not a solution.

4. Repeat the process for the second ordered triple (-3, 1, -4).

5. Set up the calculations for each equation, but stop before evaluating the final results.

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Q2. Solve the system by substitution: ,

Background

Topic: Solving Systems of Linear Equations (Substitution Method)

This question tests your ability to solve a system of two equations using substitution.

Key Terms and Formulas

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

Step-by-Step Guidance

1. Solve the second equation for :

2. Express in terms of and substitute into the first equation.

3. Solve for in the resulting equation.

4. Once you have , substitute back to find .

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Q3. Solve the system of equations by the elimination method: ,

Background

Topic: Solving Systems of Linear Equations (Elimination Method)

This question tests your ability to use elimination to solve a system of two equations.

Key Terms and Formulas

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

Step-by-Step Guidance

1. Multiply one or both equations as needed so the coefficients of or are opposites.

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

3. Solve for the remaining variable.

4. Substitute back to find the other variable.

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Q4. Use the substitution or elimination method to solve the system:

Background

Topic: Solving Systems of Linear Equations with Fractions

This question tests your ability to handle equations with fractional coefficients using substitution or elimination.

Key Terms and Formulas

• Fractional coefficients: Coefficients that are fractions, requiring careful arithmetic.

Step-by-Step Guidance

1. Clear fractions by multiplying both sides of each equation by the least common denominator (LCD).

2. Rewrite the system with integer coefficients.

3. Use substitution or elimination to solve for one variable.

4. Substitute back to find the other variable.

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Q5. Use a system of equations to solve the following problem:

On a certain hot summer's day, 572 people used the public swimming pool. The daily prices are $1.75 for children and $2.25 for adults. The receipts for admission totaled $1144.00. How many children and how many adults swam at the public pool that day?

Background

Topic: Systems of Equations – Word Problems

This question tests your ability to translate a word problem into a system of equations and solve for two unknowns.

Key Terms and Formulas

• Let = number of children, = number of adults.

• System:

Step-by-Step Guidance

1. Write the two equations based on the information given.

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

3. Substitute into the second equation and solve for the remaining variable.

4. Substitute back to find the other variable.

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Q6. Write a system of linear equations in three variables and then use matrices to solve the system using Gaussian elimination or Gauss-Jordan elimination.

Ben was in charge of ordering 30 pizzas for the office party. He ordered three types of pizza: Cheese ($7 each), Pepperoni ($10 each), and Supreme ($13 each). He spent exactly twice as much on the pepperoni pizzas as he did on the cheese pizzas. If Ben spent a total of $288 on pizza, how many pizzas of each type did he buy?

Background

Topic: Systems of Equations in Three Variables; Matrices

This question tests your ability to set up and solve a system of three equations using matrices and elimination methods.

Key Terms and Formulas

• Let = cheese, = pepperoni, = supreme.

• System:

  • (or )

Step-by-Step Guidance

1. Write the three equations based on the problem statement.

2. Express one variable in terms of another using the "twice as much" condition.

3. Substitute into the other equations to reduce the system to two variables.

4. Set up the augmented matrix for the system and begin Gaussian elimination.

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Q7. Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form.

Background

Topic: Gaussian Elimination for Systems of Three Equations

This question tests your ability to use Gaussian elimination to solve a system of three equations.

Key Terms and Formulas

• Gaussian elimination: A method for solving systems by transforming the system into upper triangular form.

Step-by-Step Guidance

1. Write the system as an augmented matrix.

2. Use row operations to create zeros below the leading 1 in the first column.

3. Continue row operations to get zeros below the leading 1 in the second column.

4. Back-substitute to solve for each variable.

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Q8. Use Gauss-Jordan elimination to solve the linear system and determine whether the system has a unique solution, no solution, or an infinite number of solutions.

Background

Topic: Gauss-Jordan Elimination

This question tests your ability to use Gauss-Jordan elimination to solve a system and classify the solution type.

Key Terms and Formulas

• Gauss-Jordan elimination: A method for reducing a system to reduced row-echelon form.

Step-by-Step Guidance

1. Write the system as an augmented matrix.

2. Use row operations to get leading 1s and zeros elsewhere in each column.

3. Interpret the final matrix to determine if there is a unique solution, no solution, or infinitely many solutions.

4. If infinite, express the solution in terms of a free variable.

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Q9. The augmented matrix in row reduced form is equivalent to the augmented matrix of a system of linear equations in variables x, y, and z. Determine whether the system is dependent or inconsistent. If dependent, determine which variable is free and describe the solution as an ordered triple in terms of a free variable.

Matrix:

Background

Topic: Interpreting Row-Reduced Matrices

This question tests your ability to interpret the solution set of a system from its row-reduced matrix form.

Key Terms and Formulas

• Dependent system: Infinitely many solutions, with at least one free variable.

• Inconsistent system: No solution.

Step-by-Step Guidance

1. Analyze the last row to check for inconsistency (e.g., means no solution).

2. If the last row is all zeros, the system is dependent.

3. Identify which variable is free (the one without a leading 1 in its column).

4. Express the solution in terms of the free variable.

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Q10. Determine the quadratic function whose graph passes through the three points (0, 0), (−5, 25), and (5, 75).

Background

Topic: Quadratic Functions and Systems of Equations

This question tests your ability to find a quadratic function given three points by setting up and solving a system of equations.

Key Terms and Formulas

• Quadratic function:

Step-by-Step Guidance

1. Substitute each point into to get three equations.

2. Solve the system for , , and .

3. Write the final quadratic function.

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Q11. Determine if each ordered pair is a solution to the given system of linear inequalities in two variables.

Background

Topic: Systems of Linear Inequalities

This question tests your ability to check if a point satisfies both inequalities.

Key Terms and Formulas

• Ordered pair:

• System of inequalities: Multiple inequalities that must be satisfied simultaneously.

Step-by-Step Guidance

1. For each ordered pair, substitute and into both inequalities.

2. Check if both inequalities are true for each pair.

3. Repeat for all given pairs.

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Q12. Graph the following inequality:

Background

Topic: Graphing Linear Inequalities

This question tests your ability to graph a linear inequality and shade the correct region.

Key Terms and Formulas

• Linear inequality: An inequality involving a linear function.

Step-by-Step Guidance

1. Graph the boundary line (solid line since ).

2. Shade the region below the line to represent .

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Q13. Graph the following inequality:

Background

Topic: Graphing Linear Inequalities

This question tests your ability to graph a strict linear inequality.

Key Terms and Formulas

• Strict inequality: Use a dashed line for or .

Step-by-Step Guidance

1. Graph the boundary line (dashed line since ).

2. Shade the region below the line to represent .

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Q14. Graph the following system of linear inequalities in two variables: ,

Background

Topic: Graphing Systems of Linear Inequalities

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

Key Terms and Formulas

• Graph each boundary line and shade the appropriate region for each inequality.

Step-by-Step Guidance

1. Rewrite each inequality in slope-intercept form if needed.

2. Graph each line (solid since ) and shade the correct side for each.

3. The solution is the overlapping (intersection) region.

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Q15. Graph the solution set of the following system of inequalities: ,

Background

Topic: Graphing Systems of Linear and Quadratic Inequalities

This question tests your ability to graph a quadratic and a linear inequality and find their intersection.

Key Terms and Formulas

• Graph the parabola (solid line, shade above).

• Graph the line (dashed line, shade below).

Step-by-Step Guidance

1. Graph both boundaries on the same axes.

2. Shade the region above the parabola and below the line.

3. The solution is the overlapping region.

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Q16. Use matrices A and B to find A + B.

Background

Topic: Matrix Addition

This question tests your ability to add two matrices of the same size by adding corresponding elements.

Key Terms and Formulas

• Matrix addition:

Step-by-Step Guidance

1. Write out matrices A and B.

2. Add corresponding elements to form the new matrix.

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Q17. Let . Find .

Background

Topic: Scalar Multiplication of Matrices

This question tests your ability to multiply a matrix by a scalar.

Key Terms and Formulas

• Scalar multiplication:

Step-by-Step Guidance

1. Multiply each entry in matrix B by 5.

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Q18. Use matrices A, B, and C to find AB + BC.

Background

Topic: Matrix Multiplication and Addition

This question tests your ability to multiply matrices and add the results.

Key Terms and Formulas

• Matrix multiplication: where

Step-by-Step Guidance

1. Multiply matrices A and B to get AB.

2. Multiply matrices B and C to get BC.

3. Add the resulting matrices.

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Q19. The youth from a local church are having a breakfast fund-raising event. They are planning on serving biscuits, pancakes, and waffles. The ingredients for one batch of each and their requirements are given by matrices A and B.

Background

Topic: Matrix Multiplication in Applications

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

Key Terms and Formulas

• Matrix multiplication: Used to calculate total ingredient requirements.

Step-by-Step Guidance

1. Multiply matrix B (batches) by matrix A (ingredients per batch) to get total ingredients needed.

2. Interpret each entry in the resulting matrix as the total amount of each ingredient required.

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