In Exercises 5–18, solve each system by the substitution method. 2x - 3y = 8 - 2x 3x + 4y = x + 3y + 14

Verified step by step guidance

1. Step 1: Simplify each equation to standard form. For the first equation, combine like terms: $\mathrm{2x}-3y=8-2x$ becomes $\mathrm{2x}+\mathrm{2x}-3y=8$, which simplifies to $4x-3y=8$.
2. Step 2: Simplify the second equation similarly: $3x+4y=x+3y+14$ becomes $3x-x+4y-3y=14$, which simplifies to $2x+y=14$.
3. Step 3: Solve one of the equations for one variable. For example, solve the second equation for $y$: $y=14-2x$.
4. Step 4: Substitute the expression for $y$ from Step 3 into the first equation: $4x-3(14-2x)\; =8$.
5. Step 5: Simplify and solve the resulting equation for $x$. Then, substitute the value of $x$ back into the expression for $y$ to find the value of $y$.

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. The solution is the set of variable values that satisfy all equations simultaneously. Understanding how to interpret and manipulate these equations is essential for solving the system.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, making it easier to solve. It is particularly useful when one variable is easily isolated.

Simplifying and Rearranging Equations

Before applying substitution, equations often need to be simplified and rearranged to isolate variables or combine like terms. This step ensures clarity and accuracy in substitution and helps avoid errors during solving. Mastery of algebraic manipulation is crucial here.