Problem 87 - Algebra A Combined Function

Question

Solve. $$ 1000(7 x-10)=50(412+100 x) $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 15.3 \).

Step 1: Expand both sides

First, expand both sides of the equation to eliminate the parentheses. For the left side: \[ 1000(7x - 10) = 7000x - 10000 \]For the right side: \[ 50(412 + 100x) = 20600 + 5000x \]

Step 2: Get all terms with x on one side

To simplify the equation, move all terms with \(x\) to one side of the equation and constant terms to the other.\[ 7000x - 5000x = 20600 + 10000 \]

Step 3: Simplify the equation

Subtract \(5000x\) from \(7000x\) and add \(10000\) to \(20600\):\[ 2000x = 30600 \]

Step 4: Solve for x

Divide both sides by 2000 to isolate \(x\):\[ x = \frac{30600}{2000} \]Simplifying gives:\[ x = 15.3 \]

Key Concepts

Algebra

Algebra is all about finding the unknown by using letters or symbols to represent numbers. It's like a math language that helps us solve problems with equations. In this exercise, we use an equation where \(x\) is the unknown number we're trying to find. Algebra lets us set up relationships between different quantities, like what's given and what's unknown. This is done using the equal sign and various operations like addition, subtraction, multiplication, and division. When working with algebra, remember these key points:

• Identify what's given and what's unknown.
• Translate words into math symbols and equations.
• Always aim to isolate the variable, which makes solving the equations easier.

Equation Expansion

Equation expansion, also known as distributing, involves getting rid of parentheses by multiplying each term inside them by the number outside. For this problem, expansion is our first step.On the left side of the equation, we're multiplying \(1000\) by everything inside the parentheses \((7x - 10)\). This gives:\[ 7000x - 10000 \]On the right side, \(50\) is multiplying \((412 + 100x)\). This results in:\[ 20600 + 5000x \]Expansion helps us simplify the equation by removing parentheses, making the problem look more straightforward. Once expanded, each side of the equation becomes easier to handle when getting ready to solve for \(x\).

Isolation of Variable

Next comes the process of isolating the variable, which means getting \(x\) all by itself on one side of the equation. This involves a few strategic moves to simplify the equation until all you're left with is \(x\).From our expanded equation:\[ 7000x - 10000 = 20600 + 5000x \]To get \(x\) on one side, subtract \(5000x\) from both the left and right sides. This pushes all terms with \(x\) on the same side:\[ 7000x - 5000x = 20600 + 10000 \]After performing this operation, you're one step closer to solving for \(x\). By isolating \(x\), we streamline the problem, making further simplification much more straightforward.

Equation Simplification

Equation simplification means reducing the equation until it's as straightforward as possible. By simplifying, we refine the equation's form, making it easy to solve for \(x\).After isolating \(x\) terms, we had\[ 2000x = 30600 \]All that is left to do now is solve for \(x\) by dividing both sides by \(2000\). Simplification at this step delivers:\[ x = \frac{30600}{2000} \]This fraction further simplifies to:\[ x = 15.3 \]Through simplification, we've managed to find that \(x\) is equal to \(15.3\). Each operation brings us closer to the neat and tidy solution, showing that sometimes math requires breaking things down step by step.