Solve each system by substitution.
3y = 5x + 6
x + y = 2

Verified step by step guidance

1. Start with the given system of equations: \$3y = 5x + 6\( and \)x + y = 2$.
2. Solve one of the equations for one variable. For example, solve the second equation for \(y\): \(y = 2 - x\).
3. Substitute the expression for \(y\) from the second equation into the first equation: \$3(2 - x) = 5x + 6$.
4. Simplify and solve the resulting equation for \(x\): distribute the 3 and combine like terms to isolate \(x\).
5. Once you find the value of \(x\), substitute it back into \(y = 2 - x\) to find the corresponding 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 goal is to find values for the variables that satisfy all equations simultaneously. In this problem, the system involves two equations with variables x and y.

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. After finding one variable, substitute back to find the other.

Solving Linear Equations

Solving linear equations means isolating the variable to find its value. This often involves operations like addition, subtraction, multiplication, division, and simplifying expressions. Accurate manipulation is essential to correctly solve for variables in the system.