Problem 16 - College Algebra

Question

In Exercises \(5-18\), solve each system by the substitution method. $$ \begin{aligned} &3 x-4 y=x-y+4\\\ &2 x+6 y=5 y-4 \end{aligned} $$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 2\) and \(y = 0\).

Step 1: Simplify the Equations

Simplify the two equations to look clearer. The first equation simplifies to \(2x + 4y = 4\), while the second one simplifies to \(2x + y = 4\).

Step 2: Solve the First Equation for \(x\)

Isolate \(x\) in the first equation. You can subtract \(4y\) from both sides to get \(x = 2 - 2y\).

Step 3: Substitute Into the Second Equation

Substitute \(x = 2 - 2y\) from the first equation into the second equation. This gives \(2(2 - 2y) + y = 4\). Simplify it to \(4 - 4y + y = 4\), which becomes \(-3y = 0\). Solve this for \(y\) to get \(y = 0\).

Step 4: Substitute \(y = 0\) Back into the First Equation

Substitute \(y = 0\) into the first equation to find \(x\). The first equation then becomes \(2x + 4(0) = 4\), which simplifies to \(2x = 4\) and then \(x = 2\).

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. In algebra, we often encounter systems with linear equations, which graphically represent straight lines. The goal when solving a system of equations is to find the value of the variables that satisfy all the equations in the system simultaneously.

Using the substitution method, one equation in the system is rearranged to express one variable in terms of the others. This expression is then substituted into the other equation(s), making it possible to solve for the remaining variables. The substitution method is particularly efficient when one of the equations can be easily solved for one of the variables, as seen in the textbook exercise.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Variables, which are represented by letters, can stand for unknown values that we aim to solve for.

In solving our provided textbook exercise, simplifying the algebraic expressions is a crucial first step. This might involve combining like terms (terms that have the same variables raised to the same power) or rearranging the terms to isolate one variable. Understanding how to manipulate these expressions is essential in the substitution method when one variable is expressed as an algebraic equation of the other.

Solving Linear Equations

Solving linear equations is a fundamental aspect of algebra that involves finding the value of the unknown variables. A linear equation is an equation where the variable(s) are not raised to any power other than one. The solutions to these equations are where their graphs intersect on the coordinate plane.

In the context of the textbook example, after simplification and substitution, the result is a simplified linear equation that can be solved for the remaining variable. Once one variable is found, it's crucial to substitute that solution back into one of the original equations to solve for the other variable, thus providing the solution to the system.