Problem 10 - College Algebra Essentials

Question

In Exercises \(5-18,\) solve each system by the substitution method. $$ \left\\{\begin{array}{l} x=3 y+7 \\ x=2 y-1 \end{array}\right. $$

Step-by-Step Solution

Answer

So the solution to the system of equations is x=-17 and y=-8

Step 1: Setup

We are given the system of equations \[ \left\{ \begin{array}{l} x=3y+7 \ x=2y-1 \end{array} \right. \]

Step 2: Substitution

Since both equations are already isolated for x, we can set them equal to each other to get: \[3y + 7 = 2y - 1\]

Step 3: Simplify

By rearranging the equation, we find that y equals -8. \[ 3y - 2y = -1 -7 \Rightarrow y = -8 \]

Step 4: Substitute Back

Substitute y=-8 into the first equation, x=3y+7, we can find x. \[ x = 3(-8)+ 7 = -17 \]

Key Concepts

System of Equations

Understanding how to solve a system of equations is a crucial skill in algebra. This involves finding the values of variables that satisfy more than one equation simultaneously. In the given problem, we're asked to find the values of variables x and y that work for both equations:

• \( x = 3y + 7 \)
• \( x = 2y - 1 \)

The substitution method, one of several techniques for solving systems, involves isolating one variable in one of the equations and then substituting the expression for this variable into the other equation. What's key here is that for every pair of values for x and y, they must make both equations true at the same time.

It's like finding the intersection point on a graph where the lines described by each equation would cross. This point represents the solution to the system, meaning both x and y values work in both equations.

Algebraic Manipulation

Algebraic manipulation is about strategically applying mathematical operations to rearrange and simplify equations. When we set the expressions for x in both equations equal to each other, as seen in the provided exercise solution, we're using algebraic manipulation to pave the way for finding the value of y. By subtracting 2y and adding 1 to both sides of the equation \( 3y + 7 = 2y - 1 \), we end up with \( y = -8 \), which simplifies the problem considerably.

Why Algebraic Manipulation is Important

It allows us to break down complex problems into simpler parts that are more easily solved. In our textbook problem, algebraic manipulation brings us to a place where we only have to deal with one variable at a time, making the solution process much more manageable.

Simplifying Equations

Simplifying equations reduces them to their most elementary form, making it easier to interpret the solution. The process includes combining like terms, reducing fractions, and clearing parentheses. In the context of our exercise, after using algebraic manipulation to deduce that \( y = -8 \), we reach the simplification stage.

Substituting and Simplifying

When we substitute \( y = -8 \) back into the first equation, x becomes \( x = 3(-8) + 7 \), which simplifies to \( x = -17 \). Simplifying the equation not only helps in finding the solution but also in verifying it.

Through simplification, students can efficiently check their results by substituting the found values of x and y back into the original equations to ensure they make true statements. If both original equations are satisfied, it verifies the correctness of the solution.