In Exercises 99–106, solve each equation. 0.7x + 0.4(20) = 0.5(x + 20)

Verified step by step guidance

1. Distribute the constants to simplify the equation. For the term 0.4(20), multiply 0.4 by 20. Similarly, distribute 0.5 to both terms inside the parentheses in 0.5(x + 20).
2. Rewrite the equation after distribution. It will look like: 0.7x + 8 = 0.5x + 10.
3. Isolate the variable term on one side of the equation. Subtract 0.5x from both sides to get: 0.7x - 0.5x + 8 = 10.
4. Simplify the coefficients of x on the left-hand side. Combine like terms to get: 0.2x + 8 = 10.
5. Isolate x by subtracting 8 from both sides, then divide by 0.2 to solve for x. The equation will become: x = (10 - 8) / 0.2.

Key Concepts

Linear Equations

A linear equation is an algebraic expression that represents a straight line when graphed. It typically takes the form ax + b = c, where a, b, and c are constants. Understanding how to manipulate and solve these equations is fundamental in algebra, as it involves isolating the variable to find its value.

Distributive Property

The distributive property states that a(b + c) = ab + ac, allowing us to multiply a single term by two or more terms inside a set of parentheses. This property is essential for simplifying expressions and solving equations, as it helps eliminate parentheses and combine like terms effectively.

Combining Like Terms

Combining like terms involves simplifying an expression by adding or subtracting terms that have the same variable raised to the same power. This process is crucial in solving equations, as it helps to reduce the complexity of the equation, making it easier to isolate the variable and find its value.