Find all values of x satisfying the given conditions. y₁ = 5(2x - 8) - 2, y₂ = 5(x - 3) + 3, and y₁ = y₂.

Verified step by step guidance

1. Start by setting the two equations for y1 and y2 equal to each other, since y1 = y2. This gives the equation: 5(2x - 8) - 2 = 5(x - 3) + 3.
2. Distribute the 5 on both sides of the equation. For the left-hand side, distribute 5 to (2x - 8), and for the right-hand side, distribute 5 to (x - 3). This results in: 10x - 40 - 2 = 5x - 15 + 3.
3. Simplify both sides of the equation by combining like terms. On the left-hand side, combine -40 and -2 to get -42. On the right-hand side, combine -15 and 3 to get -12. The equation now becomes: 10x - 42 = 5x - 12.
4. Isolate the variable x by first eliminating the 5x term from the right-hand side. Subtract 5x from both sides: 10x - 5x - 42 = -12. This simplifies to: 5x - 42 = -12.
5. Solve for x by adding 42 to both sides to isolate the term with x: 5x = 30. Then divide both sides by 5 to solve for x: x = 6.

Key Concepts

Linear Equations

Linear equations are mathematical statements that express the equality of two linear expressions. They can be represented in the form y = mx + b, where m is the slope and b is the y-intercept. Understanding how to manipulate and solve these equations is crucial for finding the values of x that satisfy the given conditions.

Substitution Method

The substitution method involves replacing one variable with an equivalent expression from another equation. In this context, setting y1 equal to y2 allows us to create a single equation in terms of x. This method simplifies the problem and makes it easier to solve for the unknown variable.

Solving for x

Solving for x involves isolating the variable on one side of the equation to find its value. This process may include combining like terms, applying inverse operations, and simplifying expressions. Mastery of this concept is essential for determining the specific values of x that satisfy the equality of the two linear equations.