Solve each system by substitution.
8x - 10y = -22
3x + y = 6

Verified step by step guidance

1. Start with the given system of equations: \[8x - 10y = -22\] \[3x + y = 6\]
2. Solve the second equation for \(y\) in terms of \(x\): \[3x + y = 6 \implies y = 6 - 3x\]
3. Substitute the expression for \(y\) from step 2 into the first equation: \[8x - 10(6 - 3x) = -22\]
4. Distribute the \(-10\) and simplify the equation: \[8x - 60 + 30x = -22\] Combine like terms to get a single-variable equation in \(x\).
5. Solve the simplified equation for \(x\), then substitute the value of \(x\) back into the expression for \(y\) from step 2 to find \(y\).

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. Understanding how these equations represent lines and their intersections is fundamental to 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 especially useful when one equation is easily solved for one variable.

Solving Linear Equations

Solving linear equations means isolating the variable to find its value. This involves using inverse operations like addition, subtraction, multiplication, and division. Mastery of these techniques is essential to manipulate equations correctly during substitution.