Problem 46 - Prealgebra

Question

The following equations contain parentheses. Apply the distributive property to remove the parentheses, then simplify each side before using the addition property of equality. $$-(x-7)+2 x-8=4$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 5\).

Step 1: Apply the Distributive Property

We begin by applying the distributive property to the terms inside the parentheses. The expression \(-(x-7)\) can be rewritten as \(-x + 7\). The equation then becomes: \(-x + 7 + 2x - 8 = 4\).

Step 2: Combine Like Terms on the Left Side

Next, combine like terms on the left side of the equation. Combine \(-x\) and \(2x\): \(-x + 2x = x\). Combine the constant terms \(7\) and \(-8\): \(7 - 8 = -1\). So, the equation simplifies to: \(x - 1 = 4\).

Step 3: Solve for x Using the Addition Property of Equality

To isolate \(x\), use the addition property of equality. Add \(1\) to both sides of the equation: \(x - 1 + 1 = 4 + 1\), which simplifies to \(x = 5\).

Key Concepts

Combining Like Terms

Combining like terms is an essential skill when simplifying algebraic expressions. It involves grouping together terms that have the same variable raised to the same power. For example, consider the expression \(-x + 2x\). Both terms have the variable \(x\) raised to the same power. When you combine them, you simply add the coefficients of these terms.

• The coefficient of \(-x\) is \(-1\).
• The coefficient of \(2x\) is \(2\).
• Adding \(-1\) and \(2\) gives you \(1\).

Therefore, \(-x + 2x\) simplifies to \(x\). Similarly, constant terms can be combined. For instance, in the equation \(7 - 8\), combining these constants results in \(-1\). This step simplifies the equation and prepares it for solving.

Addition Property of Equality

The addition property of equality is an important principle used to keep an equation balanced. This property allows you to add the same value to both sides of an equation without changing its equality. It's critical in solving linear equations where you need to isolate the variable.Consider the simpler equation \(x - 1 = 4\). To solve for \(x\), you must eliminate \(-1\) from the left-hand side. You do this by adding \(1\) to both sides of the equation.

• Add \(1\) to the left: \(x - 1 + 1\) becomes \(x\).
• Add \(1\) to the right: \(4 + 1\) becomes \(5\).

Now the equation simplifies to \(x = 5\). This principle helps maintain the balance of the equation while isolating the variable.

Solving Linear Equations

Solving linear equations is a fundamental process in algebra involving finding the value of the unknown variable that makes the equation true. Using techniques like combining like terms and the addition property of equality are crucial in reaching the solution.Given the equation \(-(x-7)+2x-8=4\), after applying the distributive property and combining like terms, we simplify it to \(x - 1 = 4\). Our final step is to solve for \(x\).We isolate \(x\) by using the addition property of equality to add \(1\) to both sides, which simplifies the equation to \(x = 5\). This value of \(x\) is the solution to the equation, confirming that the left side equals the right as both evaluate to \(5\). This structured approach is key to solving linear equations reliably.