Problem 41 - 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. $$-3(x-4)+4 x=3-7$$

Step-by-Step Solution

Answer

x = -16.

Step 1: Apply the Distributive Property

Begin by applying the distributive property to the expression \(-3(x-4)\). Distribute the \(-3\) across the terms inside the parentheses: \[-3(x - 4) = -3 \cdot x + (-3) \cdot (-4) = -3x + 12.\] So, the equation becomes:\[-3x + 12 + 4x = 3 - 7.\]

Step 2: Combine Like Terms

Next, combine like terms on the left side of the equation. The terms \(-3x\) and \(4x\) can be combined:\[-3x + 4x = (4 - 3)x = x.\]This results in the equation:\[x + 12 = 3 - 7.\]

Step 3: Simplify the Right Side

Simplify the right side of the equation:Calculate \(3 - 7\): \[3 - 7 = -4.\]So the equation simplifies to:\[x + 12 = -4.\]

Step 4: Use the Addition Property of Equality

Now, use the addition property of equality to solve for \(x\). Subtract \(12\) from both sides to isolate \(x\):\[x + 12 - 12 = -4 - 12.\]This simplifies to:\[x = -16.\]

Key Concepts

Distributive Property

In prealgebra, mastering the distributive property is key to simplifying algebraic expressions that contain parentheses. The distributive property helps you "distribute" a multiplying factor across terms within parentheses.
For example, in the expression \(-3(x - 4)\), the minus three \(-3\) needs to be multiplied across each term inside the parentheses:
\[-3 \cdot x + (-3) \cdot (-4).\]
By using the distributive property, you effectively eliminate the parentheses and transform the expression into a more workable form: \[-3x + 12.\]

When you "distribute" successfully, it not only simplifies the equation but also sets up the subsequent steps seamlessly. Apply this initial step correctly to make combining like terms and solving the equation a smooth process.

Combining Like Terms

Once you have simplified an equation using the distributive property, the next step is to "combine like terms." Look for terms on the same side of the equation that have the same variable part.
For instance, consider \(-3x + 4x\). Both terms include the variable \(x\), so you can combine them:

• Add their coefficients, giving you: \((4 - 3)x\) = x.\

This consolidates the expression effectively, reducing clutter and making the equation simpler.

Combining terms not only clarifies the mathematical relationship between numbers and variables but also increases the manageability of an equation. Master this step to smooth out the solving process and focus on isolating the variable.

Addition Property of Equality

The addition property of equality is a fundamental principle that helps you solve equations by maintaining balance. It states that you can add or subtract the same amount from both sides of an equation, and the equation will remain true.

For instance, in an equation like \( x + 12 = -4 \), you can subtract the same number 12 from both sides to isolate \(x\):

• Subtract 12 on the left: \(x + 12 - 12 = x.\)
• Subtract 12 on the right: \(-4 - 12 = -16.\)

This results in \(x = -16\), and the variable \(x\) is now solved.

By applying this property correctly, you maintain the balance of the equation while making the variable the focal point. This concept ensures you can find the value of the variable in a consistent manner, completing the solution process effortlessly.