Solve each system by substitution.
-2x = 6y + 18
-29 = 5y - 3x

Verified step by step guidance

1. Start by isolating one variable in one of the equations. For example, take the first equation \(-2x = 6y + 18\) and solve for \(x\). To do this, divide both sides by \(-2\) to get \(x\) in terms of \(y\): \(x = \frac{6y + 18}{-2}\).
2. Simplify the expression for \(x\) from the first step: \(x = -3y - 9\).
3. Substitute the expression for \(x\) from step 2 into the second equation \(-29 = 5y - 3x\). Replace \(x\) with \(-3y - 9\) to get an equation with only \(y\): \(-29 = 5y - 3(-3y - 9)\).
4. Simplify the equation from step 3 by distributing the \(-3\) and combining like terms: \(-29 = 5y + 9y + 27\).
5. Solve the simplified equation for \(y\) by isolating \(y\) on one side. Once you find \(y\), substitute it back into the expression for \(x\) from step 2 to find the value of \(x\).

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 equation is easily solved for a variable.

Algebraic Manipulation

Algebraic manipulation includes operations like isolating variables, distributing, combining like terms, and simplifying expressions. These skills are necessary to rearrange equations correctly during substitution and to solve for variables accurately.