Problem 22 - College Algebra

Question

Solve each system by addition. $$ \begin{array}{l} 6 x-5 y=-34 \\ 2 x+6 y=4 \end{array} $$

Step-by-Step Solution

Answer

\( x = \frac{92}{13} \), \( y = -\frac{22}{13} \).

Step 1: Setup the system

We are given the system of equations:\[ \begin{align*} 6x - 5y &= -34 \ 2x + 6y &= 4 \end{align*} \] Our goal is to eliminate one variable by using the addition method.

Step 2: Multiply the equations to align coefficients

To use the addition method, we need the coefficients of either \( x \) or \( y \) to be equal in magnitude. Let's eliminate \( x \) by making their coefficients equal. Multiply the second equation by 3:\[ 3(2x + 6y) = 3(4) \] This gives us:\[ 6x + 18y = 12 \]

Step 3: Add the modified equations

Now add the transformed second equation to the first equation:\[ \begin{align*} (6x - 5y) + (6x + 18y) &= (-34) + 12 \ 12x + 13y &= -22 \end{align*} \]This eliminates \( x \) and gives us:\[ 13y = -22 \].

Step 4: Solve for y

Solve the new equation for \( y \):\[ \begin{align*} 13y &= -22 \ y &= \frac{-22}{13} \end{align*} \]Thus, \( y = -\frac{22}{13} \).

Step 5: Substitute back to find x

Substitute \( y = -\frac{22}{13} \) back into any original equation, let's choose the second one:\[ 2x + 6\left(-\frac{22}{13}\right) = 4 \]Simplifying gives:\[ 2x - \frac{132}{13} = 4 \]Solve for \( x \):\[ \begin{align*} 2x &= 4 + \frac{132}{13} \ 2x &= \frac{52}{13} + \frac{132}{13} \ 2x &= \frac{184}{13} \ x &= \frac{92}{13} \end{align*} \]Thus, \( x = \frac{92}{13} \).

Step 6: Final solution

The solution to the system is \( x = \frac{92}{13} \) and \( y = -\frac{22}{13} \).

Key Concepts

Addition Method

The addition method is a clever strategy for solving systems of two linear equations. This technique relies on rearranging and manipulating equations so we can eliminate one variable. When two equations are combined, and one of the variable's coefficients are aligned to cancel each other, the linear system becomes easier to solve.

Here's a simple breakdown of how the addition method works:

• First, inspect the given equations and decide which variable—either \(x\) or \(y\)—you want to eliminate.
• If the coefficients of the chosen variable aren't already opposites, multiply one or both equations by necessary factors to align them.
• Perform addition across the equations, which should result in the chosen variable being eliminated.
• With one variable gone, you'll have a single linear equation to solve with the remaining variable.

The resulting solution can be inserted back into one of the original equations to determine the other variable. This process makes the addition method a powerful, straightforward tool in linear algebra.

Elimination Technique

The elimination technique is a popular approach when solving systems of equations, particularly linear equations. This technique is synonymous with the addition method, as both strive to eliminate one variable to simplify solving the system.

To successfully apply the elimination technique, follow these practical steps:

• Start with your system of linear equations and determine which variable you want to target for elimination.
• Adjust the coefficients so that when the equations are combined, one of the variables is cancelled out. This adjustment might involve multiplying the entire equation by certain factors.
• Once ready, add or subtract the equations from each other. This step should eliminate one of the variables entirely, leaving you with an equation in a single variable.
• With one variable removed, you can easily solve for the remaining variable.
• Finally, substitute the found solution back into one of the original equations to solve for the first variable that was eliminated.

Using the elimination technique reduces the complexity of solving systems of linear equations and helps to arrive at the solution efficiently.

Linear Equations

Linear equations form the backbone of algebra, found in many real-world applications, from calculating finances to physics problems involving constant rates. These equations, characterized by their linearity, signify that variables increase or decrease in proportion to each other.

Here's what you need to know about linear equations:

• A linear equation generally looks like \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables.
• They graph as straight lines, which is a significant feature distinguishing them from quadratic or other polynomial equations that curve.
• Solving one linear equation involves finding values of variables that make the equation true, typically resulting in infinite solutions along a line.
• However, with a system of linear equations (at least two), the solutions are often a single point where the equations intersect, which aligns perfectly with the idea of solving using the addition method or the elimination technique.

Mastering linear equations is crucial for further diving into complex areas of mathematics, as they serve as building blocks for understanding higher-level algebra and calculus.