College Algebra: Systems of Equations Study Guide

Q1. Determine if each ordered pair is a solution of the system of equations:

Given the system:

• $4x + 5y = 2$

• $x - 2y = -6$

Check if the points (2,2) and (−2,2) are solutions.

Background

Topic: Systems of Linear Equations – Solutions

This question tests your ability to determine if a given ordered pair is a solution to a system of equations by substituting the values into each equation.

Key Terms and Formulas:

• Ordered Pair: A pair of numbers (x, y) that may satisfy both equations in the system.

• Solution to a System: An ordered pair that makes both equations true.

Step-by-Step Guidance

1. Substitute $x = 2$ and $y = 2$ into the first equation: $4x + 5y = 2$.

2. Check if the left side equals the right side after substitution.

3. Substitute $x = 2$ and $y = 2$ into the second equation: $x - 2y = -6$.

4. Repeat the process for the point $(-2, 2)$ in both equations.

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Q2. Solve the system of equations graphically:

• $y = 5x + 15$

• $y = -7x - 33$

Background

Topic: Graphing Systems of Linear Equations

This question tests your ability to graph two linear equations and determine their point of intersection, which represents the solution to the system.

Key Terms and Formulas:

• Slope-Intercept Form: $y = mx + b$

• Intersection Point: The (x, y) value where both lines meet.

Step-by-Step Guidance

1. Identify the slope and y-intercept for each equation.

2. Graph both lines on the same coordinate plane using their slopes and y-intercepts.

3. Observe where the two lines intersect. This point is the solution to the system.

4. If the lines are parallel, there is no solution. If they are the same line, there are infinitely many solutions.

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Q3. Does the system have a unique solution, no solution, or many solutions?

• $2x + 13y = 6$

• $x + 6.5y = 3$

Background

Topic: Types of Solutions for Linear Systems

This question tests your understanding of when a system has one solution, no solution, or infinitely many solutions, and what that means graphically.

Key Terms and Formulas:

• Unique Solution: The lines intersect at one point.

• No Solution: The lines are parallel.

• Many Solutions: The lines are coincident (the same line).

Step-by-Step Guidance

1. Compare the ratios of the coefficients of $x$ and $y$ in both equations.

2. If the ratios are equal for both $x$ and $y$ but not for the constants, the lines are parallel (no solution).

3. If all ratios are equal, the lines are the same (many solutions).

4. If the ratios are different, the lines intersect at one point (unique solution).

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Q4. Solve the system of equations by substitution:

• $x = 6y + 9$

• $5x + 7y = -29$

Background

Topic: Solving Systems by Substitution

This question tests your ability to use substitution to solve a system of linear equations.

Key Terms and Formulas:

• Substitution Method: Replace one variable in an equation with its equivalent from the other equation.

Step-by-Step Guidance

1. Since $x$ is already isolated, substitute $x = 6y + 9$ into the second equation.

2. Write $5(6y + 9) + 7y = -29$.

3. Expand and combine like terms to form an equation in $y$ only.

4. Solve for $y$.

5. Once you have $y$, substitute back into $x = 6y + 9$ to find $x$.

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Q5. Solve the system of equations by substitution:

• $3x - 8y = 22$

• $7x + 5y = 4$

Background

Topic: Solving Systems by Substitution

This question tests your ability to solve a system by isolating one variable and substituting into the other equation.

Key Terms and Formulas:

• Substitution Method: Solve one equation for one variable, then substitute into the other equation.

Step-by-Step Guidance

1. Choose one equation and solve for $x$ or $y$ (whichever is easier).

2. Substitute the expression for that variable into the other equation.

3. Simplify and solve for the remaining variable.

4. Substitute back to find the other variable.

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Q6. Solve the system of equations by elimination, if a solution exists:

• $4x - y = 31$

• $x + 5y = 13$

Background

Topic: Solving Systems by Elimination

This question tests your ability to use the elimination method to solve a system of linear 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 that the coefficients of $x$ or $y$ 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.

Try solving on your own before revealing the answer!