Solve each system by substitution. See Example 1. 4x + 3y = -13-x + y = 5

Verified step by step guidance

1. Step 1: Solve the second equation for y in terms of x. The equation is -x + y = 5. Add x to both sides to get y = x + 5.
2. Step 2: Substitute the expression for y from Step 1 into the first equation. Replace y in 4x + 3y = -13 with x + 5, resulting in 4x + 3(x + 5) = -13.
3. Step 3: Simplify the equation from Step 2. Distribute the 3 to both terms inside the parentheses: 4x + 3x + 15 = -13.
4. Step 4: Combine like terms in the equation from Step 3. Combine 4x and 3x to get 7x + 15 = -13.
5. Step 5: Solve for x by isolating it. Subtract 15 from both sides to get 7x = -28, then divide both sides by 7 to find the value of x.

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.