Problem 66 - Algebra 2

Question

Which equation is equivalent to \(4(9-3 x)=7-2(6-5 x) ?\) $$ \begin{array}{ll}{\mathbf{F}8 x=41} & {\mathbf{H} 22 x=41} \\ {\mathbf{G} 8 x=24} & {\mathbf{J} 22 x=24}\end{array} $$

Step-by-Step Solution

Answer

The equivalent equation is \(22x = 41\) (option H).

Step 1: Distribute the 4 on the left side

The equation is given as \(4(9 - 3x) = 7 - 2(6 - 5x)\). Start by distributing the 4 across the terms inside the parentheses on the left side: \(4 \times 9\) and \(4 \times (-3x)\). This results in: \(36 - 12x\).

Step 2: Distribute the -2 on the right side

Next, distribute the -2 across the terms inside the parentheses on the right: \(-2 \times 6\) and \(-2 \times (-5x)\). This results in: \(-12 + 10x\).

Step 3: Write down the simplified equation

After distribution, the equation becomes \(36 - 12x = 7 - 12 + 10x\). Simplify the right side by combining \(7 - 12\) to get \(-5\), leading to: \(36 - 12x = -5 + 10x\).

Step 4: Rearrange the equation

Rearrange the equation to get all x-terms on one side and constant terms on the other. Add \(12x\) to both sides to give: \(36 = -5 + 22x\).

Step 5: Solve for the equivalent equation

Add \(5\) to both sides to begin isolating the term with x, resulting in \(41 = 22x\). This can be rewritten as \(22x = 41\) by moving terms around.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps in simplifying expressions. It's like a mathematical version of sharing: when you have a term outside of parentheses, you distribute or "share" this term with each item inside the parentheses. Let's break this down with an example based on the original exercise.

Consider the expression \(4(9 - 3x)\). To apply the distributive property, multiply the 4 by each of the terms inside the parentheses:

• Multiply \(4\) and \(9\) to get \(36\).
• Multiply \(4\) and \(-3x\) to get \(-12x\).

This results in the expression \(36 - 12x\).

Remember, when distributing, make sure to multiply each term correctly and not miss any signs. It is especially important to pay attention to negative signs as they can change the results significantly.

Linear Equations

Linear equations are equations of the first degree, meaning they involve only terms with variables raised to the power of one. In terms of their graph, they represent straight lines. Solving linear equations involves finding the value of the variable that makes the equation true.

For example, in the simplified form of the equation from our step-by-step solution, we have \( 22x = 41 \). This is a linear equation because it involves a variable \( x \) with an exponent of one.

In solving it, our goal is to isolate \( x \) on one side of the equation. This often involves rearranging the terms—which includes adding or subtracting terms on both sides—and then dividing or multiplying to get the variable by itself. This approach keeps the equation balanced, similar to keeping a see-saw level by adjusting the weight on both sides.

Algebraic Simplification

Algebraic simplification is all about making an expression easier to work with. It involves removing unnecessary parentheses, combining like terms, and rearranging terms so the expression is as straightforward as possible.

In the original problem, we simplified the equation \(36 - 12x = 7 - 12 + 10x\) to \(36 - 12x = -5 + 10x\). This involved combining the constant terms on the right side of the equation: \(7 - 12\) reduces to \(-5\).

This process makes equations easier to solve by focusing on what's essential. Another key point is rearranging terms to gather like terms together, which provides clarity and often results in more comprehensible results.

Simplification helps pave the way for solving equations effectively, especially when dealing with more complex algebraic expressions.