Problem 49 - Algebra A Combined Function

Question

Solve. $$ 10-(2 x-4)=7-3 x $$

Step-by-Step Solution

Answer

The solution is \(x = -7\).

Step 1: Distribute and rearrange

First, distribute any negative signs through the parentheses. In this case, distribute the negative sign across the \[2x - 4\]which simplifies the expression to\[10 - 2x + 4 = 7 - 3x\].Adding the constant terms on the left side simplifies to\[14 - 2x = 7 - 3x\].

Step 2: Add/Subtract to isolate variable terms

We want to move like terms (the \(x\) terms) to one side and constant terms to the other.To eliminate the \(-3x\) from the right-hand side, add \(3x\) to both sides of the equation:\[14 - 2x + 3x = 7 - 3x + 3x\]This simplifies to\[14 + x = 7\].

Step 3: Solve for the variable

Now, isolate the variable \(x\) by subtracting 14 from both sides of the equation:\[14 + x - 14 = 7 - 14\]Which simplifies to:\[x = -7\].

Key Concepts

Distributive Property

The distributive property is a useful algebraic rule that allows us to simplify expressions in a very efficient manner. When you have a term outside parentheses multiplied by a group of terms inside, you apply the property by distributing or multiplying the term outside the parentheses by each term inside individually. In the case of our exercise, we have an expression including a negative sign before parentheses. This negative sign acts as a "multiplier".For the expression \(10 - (2x - 4)\), we distribute the negative sign:

• First, multiply \-1\ times each term inside the parentheses: \-1 \cdot 2x = -2x\.
• Then, \-1 \cdot (-4) = +4\.

When we finish distributing, the expression inside the parentheses changes to \(-2x + 4\). The overall expression becomes \(10 - 2x + 4\), which we then simplify by combining like terms to get \(14 - 2x\).It is always important to simplify expressions gradually and ensure every term is accounted for correctly to maintain accuracy in algebraic manipulation.

Solving Linear Equations

Solving linear equations is a fundamental skill in algebra, as it involves finding the value of the variable that makes the equation true. A linear equation is an equation in which the variable is raised to the first power. Our given equation is one involving such linear terms: \[14 - 2x = 7 - 3x\]To solve linear equations:

• Our first step after simplification is to rearrange the equation so all terms involving the variable are on one side, and constant terms are on the other.
• In this exercise, we add \(3x\) to both sides to eliminate it from the right side, transforming the equation to \(14 + x = 7\).

By rearranging and simplifying, we make it possible to clearly see how to approach isolating the variable. Solving equations often requires moving things around and ensuring each operation is done evenly to both sides to maintain the balance of the equation.

Variable Isolation

Variable isolation refers to the process of getting the variable alone on one side of the equation. This step is crucial because it directly provides the value of the unknown variable.Let's revisit our partially solved equation: \[14 + x = 7\]To isolate the variable \(x\):

• We subtract \(14\) from both sides. This gets rid of the \(+14\) on the left, resulting in \(x\ = 7 - 14\).
• Solving this gives us \(x\ = -7\).

Once \(x\) is isolated, we find that \(x\ = -7\) is the solution. The process of variable isolation is particularly important because only when the variable stands alone do we know that we have solved the equation correctly. It ensures the equation is balanced and our solution is verified.