In Exercises 5–18, solve each system by the substitution method. x + 3y = 8 y = 2x - 9

Verified step by step guidance

1. Identify the system of equations: $x+3y=8y=2x-9$.
2. Since the second equation is already solved for $y$, substitute $y=2x-9$ into the first equation in place of $y$.
3. Rewrite the first equation after substitution: $x+\; 3(2x-9)\; =8$.
4. Distribute the 3 across the terms inside the parentheses: $x+6x-27=8$.
5. Combine like terms and solve for $x$. After finding $x$, substitute it back into $y=2x-9$ 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 represent these systems is fundamental to solving them.

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 especially useful when one equation is already solved for a variable.

Solving Linear Equations

Solving linear equations means finding the value(s) of the variable(s) that make the equation true. This often involves isolating the variable using inverse operations like addition, subtraction, multiplication, or division. Mastery of these techniques is essential for solving systems effectively.