Solving Simultaneous Equations and Applications in College Algebra

Simultaneous Equations

Definition and Overview

Simultaneous equations are a set of equations with multiple variables that are solved together, so that the solution satisfies all equations at the same time. In College Algebra, these are typically linear equations involving two variables, such as x and y.

• Linear Equation: An equation of the form $ax + by = c$, where a, b, and c are constants.

• Simultaneous Solution: A pair of values $(x, y)$ that makes both equations true.

Methods for Solving Simultaneous Equations

There are several standard methods for solving simultaneous linear equations:

1. Substitution Method: Solve one equation for one variable and substitute into the other equation.

2. Elimination Method: Add or subtract equations to eliminate one variable, making it possible to solve for the other.

3. Graphical Method: Graph both equations and find the intersection point (not required in these questions, but useful context).

Example 1: Solving by Elimination

Given the system:

• $3x - 8y = 22$

• $x + 4y = 4$

Steps:

1. Multiply the second equation by 2 to align the coefficients of y:

2. $2(x + 4y) = 2(4) \Rightarrow 2x + 8y = 8$

3. Add the first and the new equation:

4. $(3x - 8y) + (2x + 8y) = 22 + 8$

5. $5x = 30 \Rightarrow x = 6$

6. Substitute $x = 6$ into $x + 4y = 4$:

7. $6 + 4y = 4 \Rightarrow 4y = -2 \Rightarrow y = -0.5$

Solution: $x = 6$, $y = -0.5$

Example 2: Solving by Substitution

Given the system:

• $3x - y = 22$

• $x + 2y = 5$

Steps:

1. Solve the second equation for $x$:

2. $x = 5 - 2y$

3. Substitute into the first equation:

4. $3(5 - 2y) - y = 22$

5. $15 - 6y - y = 22$

6. $15 - 7y = 22$

7. $-7y = 7 \Rightarrow y = -1$

8. Substitute $y = -1$ into $x = 5 - 2y$:

9. $x = 5 - 2(-1) = 5 + 2 = 7$

Solution: $x = 7$, $y = -1$

Example 3: More Complex Coefficients

Given the system:

• $5x - 2y = 26$

• $7x + 6y = 10$

Steps:

1. Multiply the first equation by 3 to align the coefficients of $y$:

2. $3(5x - 2y) = 3(26) \Rightarrow 15x - 6y = 78$

3. Add to the second equation:

4. $(15x - 6y) + (7x + 6y) = 78 + 10$

5. $22x = 88 \Rightarrow x = 4$

6. Substitute $x = 4$ into $5x - 2y = 26$:

7. $5(4) - 2y = 26 \Rightarrow 20 - 2y = 26 \Rightarrow -2y = 6 \Rightarrow y = -3$

Solution: $x = 4$, $y = -3$

Applications of Simultaneous Equations

Word Problems Involving Cost

Simultaneous equations are often used to solve real-world problems, such as calculating costs or quantities.

Example: Calculating Total Cost

• Esme buys x magazines at $2.45 each and y cards at $3.15 each.

• Expression for total cost:

$\text{Total cost} = 2.45x + 3.15y$

Example: Finding Quantity from Total Cost

• Esme spends $60.55 in total and buys 8 magazines. How many cards does she buy?

Set up the equation:

$2.45(8) + 3.15y = 60.55$

$19.60 + 3.15y = 60.55$

$3.15y = 60.55 - 19.60 = 40.95$

$y = \frac{40.95}{3.15} = 13$

Answer: Esme buys 13 cards.

Summary Table: Methods for Solving Simultaneous Equations

Method	Steps	When to Use
Substitution	1. Solve one equation for one variable. 2. Substitute into the other equation. 3. Solve for the second variable. 4. Back-substitute to find the first variable.	When one equation is easily solved for a variable.
Elimination	1. Multiply equations to align coefficients. 2. Add or subtract equations to eliminate a variable. 3. Solve for the remaining variable. 4. Substitute back to find the other variable.	When coefficients can be easily matched or eliminated.
Graphical	1. Graph both equations on the same axes. 2. Find the intersection point.	To visualize solutions or when exact values are not required.

Key Points to Remember

• Always check your solution by substituting values back into the original equations.

• Word problems often require translating real-world information into algebraic equations.

• Simultaneous equations can have one solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (coincident lines).