Problem 88 - Algebra A Combined Function

Question

Solve. $$ 1000(x+40)=100(16+7 x) $$

Step-by-Step Solution

Answer

The solution is \(x = -128\).

Step 1: Expand Both Sides

Start by expanding both sides of the equation. Multiply the numbers outside the parentheses with the terms inside the parentheses. On the left side: \[ 1000(x + 40) = 1000x + 40000 \]On the right side:\[ 100(16 + 7x) = 1600 + 700x \]

Step 2: Rearrange Equation

Rearrange the equation so that all terms containing the variable \(x\) are on one side and the constant terms are on the other side.Bring terms involving \(x\) to one side:\[ 1000x - 700x = 1600 - 40000 \]

Step 3: Simplify

Combine like terms to simplify the equation.\[ 300x = -38400 \]

Step 4: Solve for \(x\)

Divide both sides of the equation by the coefficient of \(x\) to solve for \(x\).\[ x = \frac{-38400}{300} \]Simplify the fraction:\[ x = -128 \]

Key Concepts

Expansion of Expressions

Expansion of expressions is the process of removing parentheses by multiplying the terms outside the parentheses with every term inside. This is crucial for simplifying and solving algebraic equations, as it allows us to transform expressions into more manageable forms. For example, in the given exercise, we had the expression \(1000(x+40)\) on the left side of the equation. To expand, we multiply 1000 by each term within the parentheses.

• \(1000 \times x = 1000x\)
• \(1000 \times 40 = 40000\)

After expanding, the expression becomes \(1000x + 40000\). Similarly, the right side \(100(16 + 7x)\) expands into \(1600 + 700x\). This step lays the groundwork for further manipulation of the equation by simplifying complex expressions into simpler, linear ones.

Rearranging Equations

Rearranging equations involves shifting all terms with the variable to one side of the equation and constant terms to the other. This is vital when isolating terms to simplify solving for the unknown variable. In the given exercise, once both sides were expanded, the equation was:\[1000x + 40000 = 1600 + 700x\]The next step was to move all terms with \(x\) to one side by subtracting \(700x\) from both sides:\[1000x - 700x = 1600 - 40000\]This rearrangement makes it easier to solve for \(x\) by clearly separating terms on either side. The goal is to end up with the variable on one side and a constant on the other, setting the stage for solving the equation.

Solving Linear Equations

Solving linear equations is the process of finding the value of an unknown that makes the equation true. After rearranging the equation and simplifying, we find ourselves ready to solve for the variable. From the rearranged equation:\[300x = -38400\]The variable \(x\) can be isolated by dividing both sides of the equation by 300, the coefficient of \(x\). This operation gives:\[x = \frac{-38400}{300}\]Simplifying this fraction results in \(x = -128\), the solution to the linear equation. This process highlights the clarity and simplicity of rearranging terms to facilitate solving.

Combining Like Terms

Combining like terms simplifies an equation by reducing it to fewer terms, making it easier to solve. In this process, terms with the same variable (and exponent) are added or subtracted together. This is essential for simplifying linear equations to isolate the variable.In the context of the given example, after expanding and rearranging, the equation became:\[1000x - 700x = 1600 - 40000\]Here, \(1000x\) and \(700x\) are like terms because they both have the variable \(x\) to the same power. They are combined by subtraction:\[300x\]Similarly, the constant terms 1600 and 40000 are combined:\[1600 - 40000 = -38400\]Combining terms condenses the equation into a simpler form, \(300x = -38400\), paving the way for easier resolution by solving for \(x\). This practice is a core tool in algebra for streamlining solving processes.