Problem 5 - Algebra 1: Concepts and Skills

Question

use the following system of equations. $$ \begin{aligned} &3 x+2 y=7\\\ &5 x-y=3 \end{aligned} $$ Substitute the expression for y into the other equation and solve for x.

Step-by-Step Solution

Answer

The solution for the variable x in the system of equations is \(x = 1\)

Step 1: Solve the second equation for y

First, rearrange the second equation \(5x - y = 3\) to solve for y. Do this by subtracting \(5x\) from both sides of the equation to get \(y = 5x - 3\)

Step 2: Substitute for y in the first equation

Next, substitute the expression for \(y\) from Step 1 into the first equation. This gives us \(3x + 2(5x - 3) = 7\). Simplify this to \(3x + 10x - 6 = 7\) and then to \(13x - 6 = 7\)

Step 3: Solve for x

Now add \(6\) to both sides of the equation to isolate \(x\). This gives us \(13x = 13\). Divide both sides of the equation by \(13\) to get the final answer \(x = 1\)

Key Concepts

Substitution Method

Understanding the substitution method is crucial when solving systems of equations. It involves replacing one variable with an equivalent expression from another equation within the system. The goal is to reduce the system to a single equation with one variable, which can then be solved easily.

Let's use the provided system of equations as an example. First, we find an expression for one variable from one of the equations, in this case, solving for y in the equation, \(5x - y = 3\)). Once we have y expressed in terms of x, in this case \(y = 5x - 3\), we substitute this expression into the other equation. The substitution method transforms a system of equations into a single variable problem, streamlining the path to finding the solution.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They encapsulate relationships and values without using equality signs, different from equations which assert equality. When we solve for a variable, we are essentially manipulating these expressions to isolate the variable of interest.

In the provided exercise, \(y = 5x - 3\)) is an algebraic expression obtained by isolating y in the second equation. We then use this algebraic expression to substitute y in the first equation, intertwining the concept of algebraic expressions with the substitution method. Recognizing how to work with these expressions is fundamental to simplifying and solving equations.

Isolate Variables

To isolate a variable means to rearrange an equation so that the variable of interest is by itself on one side, with all other terms on the opposite side. The process typically involves basic arithmetic operations: addition, subtraction, multiplication, and division.

In step 1 of solving the system, we isolated y by moving the term involving x to the other side. Steps like these reduce complexity and set the stage for substitution. When we reached the point of \(13x - 6 = 7\)), the next step was to add 6 to both sides, thereby isolating x, and giving us an easily solvable equation \(13x = 13\)), leading to the solution \(x = 1\)). This skill is a cornerstone of algebra that makes solving equations more straightforward.