Problem 32 - A Graphical Approach to College Algebra

Question

Solve each system by elimination. $$\begin{array}{r}5 x+7 y=6 \\\10 x-3 y=46\end{array}$$

Step-by-Step Solution

Answer

The solution is \(x = 4\) and \(y = -2\).

Step 1: Align Equations

Write the system of equations in a column format for clarity:1. \(5x + 7y = 6\)2. \(10x - 3y = 46\).

Step 2: Equalize Coefficients of x

Multiply the first equation by 2 so that the coefficients of \(x\) in both equations match:\[2(5x + 7y) = 2(6)\]This gives us the new equation:\(10x + 14y = 12\).

Step 3: Subtract Equations to Eliminate x

Subtract the second equation from this new equation to eliminate \(x\):\[(10x + 14y) - (10x - 3y) = 12 - 46 \]Resulting in:\(17y = -34\).

Step 4: Solve for y

Divide both sides of the equation \(17y = -34\) by 17 to solve for \(y\):\[y = \frac{-34}{17} = -2\].

Step 5: Substitute y back into one of the original equations

Use \(y = -2\) in the first original equation \(5x + 7y = 6\):\[5x + 7(-2) = 6\]Simplify to find \(x\):\[5x - 14 = 6\].

Step 6: Solve for x

Add 14 to both sides to isolate \(5x\):\[5x = 20\]Then divide by 5:\[x = \frac{20}{5} = 4\].

Step 7: Solution

The solution to the system of equations is \(x = 4\) and \(y = -2\).

Key Concepts

Elimination Method

The elimination method is a popular technique used to solve systems of equations. Its primary goal is to eliminate one variable so that you can easily solve for the other. Using this method involves combining equations strategically. You can do this by either addition or subtraction once the coefficients of the variables are aligned.

Here's a quick recap:

• Align Equations: Start by writing the system in a column format. This helps you visually track the coefficients and corresponding terms.
• Matching Coefficients: Adjust the coefficients of one variable so they match, letting you eliminate it by subtraction or addition.
• Eliminate and Solve: Once a variable is eliminated, solve the resulting simple equation for the remaining variable.

By following these steps, you'll find the values for both variables efficiently. It's like peeling the layers of an onion until you're left with the core—the solution!

Algebra

Algebra is the branch of mathematics that deals with symbols and the rules for manipulating those symbols. It's foundational in high school mathematics and involves working with equations, like our system of equations exercise.

In a system of equations:

• Expressions: Algebra involves expressions that can include constants, variables, and operations (like plus or minus).
• Equations: Equations are statements of equality with expressions on both sides. Solving them involves finding the values for variables that satisfy this equality.
• Manipulations: Algebra teaches us how to rearrange and simplify these expressions, making the unknown variables easier to solve for.

Think of algebra as a toolbox that equips you with the techniques needed to crack any math puzzle. By mastering algebra, you transform complex problems into manageable solutions.

Solving Equations

Solving equations is a vital skill, and one of the core concepts in algebra. It allows you to find the unknown values that make an equation true. Here's a breakdown of the process as demonstrated in our exercise:

• Transforming Equations: Begin by transforming equations by multiplying or adding to align terms. This makes them easier to handle.
• Isolating Variables: Use strategic operations to isolate the variable. This might involve adding, subtracting, multiplying, or dividing both sides by the same number.
• Substitution: Once you solve for one variable, substitute it into another equation. This leads you to solve for the remaining variable.

These steps form a solid framework for tackling various math problems. With practice, solving equations becomes an intuitive and streamlined process, paving the way for understanding more complex mathematical concepts.