Solve each system by substitution.
7x - y = -10
3y - x = 10

Verified step by step guidance

1. Start with the given system of equations: \$7x - y = -10\( \)3y - x = 10$.
2. Solve one of the equations for one variable in terms of the other. For example, from the first equation, solve for \(y\): \(7x - y = -10 \implies y = 7x + 10\).
3. Substitute the expression for \(y\) from step 2 into the second equation: \$3(7x + 10) - x = 10$.
4. Simplify and solve the resulting equation for \(x\): \$21x + 30 - x = 10\( which simplifies to \)20x + 30 = 10$.
5. Once you find \(x\), substitute it back into the expression for \(y\) from step 2 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 variables. The goal is to find values for the variables 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 especially useful when one variable is easily isolated.

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 crucial for solving systems effectively.