Solve each system by substitution. See Example 1.8x - 10y = -22 3x + y = 6

Verified step by step guidance

1. Start by solving the second equation for y: \( y = 6 - 3x \).
2. Substitute \( y = 6 - 3x \) into the first equation: \( 8x - 10(6 - 3x) = -22 \).
3. Distribute the -10 in the equation: \( 8x - 60 + 30x = -22 \).
4. Combine like terms: \( 38x - 60 = -22 \).
5. Solve for x by adding 60 to both sides and then dividing by 38.

Key Concepts

Systems of Equations

A system of equations consists of two or more equations with the same variables. The solution to the system is the set of values that satisfy all equations simultaneously. In this case, we have two linear equations in two variables, x and y, which can be solved using various methods, including substitution.

Substitution Method

The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation. This method simplifies the system into a single equation with one variable, making it easier to find the values of both variables step by step.

Linear Equations

Linear equations are mathematical statements that represent straight lines when graphed on a coordinate plane. They can be expressed in the form Ax + By = C, where A, B, and C are constants. Understanding the properties of linear equations is crucial for solving systems, as it helps in visualizing the solutions as points of intersection between lines.