Problem 43 - Introductory Algebra for College Students

Question

Solve each equation for \(x .\) $$y=(a+b) x$$

Step-by-Step Solution

Answer

\(x = y/(a+b)\)

Step 1: Identify the Variable to Isolate

In this equation, the variable to solve for is \(x\). This means that the goal is to isolate \(x\) on one side of the equation.

Step 2: Separate terms with \(x\)

Looking at the equation, we can see that \(x\) is being multiplied by \((a+b)\). To isolate \(x\), divide both sides of the equation by \((a+b)\).

Step 3: Divide and Simplify

When carrying out the division, we obtain \(x= y/(a+b)\). Therefore, \(x\) is equal to the ratio of \(y\) to the sum of \(a\) and \(b\).

Key Concepts

Understanding Variables

Variables are symbols that stand for unknown values in an equation. In this exercise, our variable is \(x\). It's the value we want to find. Variables allow us to represent mathematical relationships clearly and concisely.
For instance, in the equation \(y = (a+b)x\), \(x\) is the variable because it's the quantity that can change depending on the values of \(a\), \(b\), and \(y\).
- Variables are usually represented by letters such as \(x\), \(y\), \(a\), \(b\), etc. - They enable us to write equations that can be solved for various different scenarios.- Identifying which letters are variables is crucial to start solving any equation.
In this problem, isolating the variable \(x\) means getting it alone on one side of the equation so we can find out its value.

The Art of Isolation

Isolation is a fundamental technique used in solving equations. The goal is to rearrange the equation so that the variable of interest—here, \(x\)—stands alone on one side.
- To achieve this, we often need to perform operations like addition, subtraction, multiplication, or division on both sides of the equation.- In our example, since \(x\) is multiplied by \((a+b)\), we need to divide both sides by \((a+b)\) to isolate \(x\).
Keeping the equation balanced is key. This means whatever operation you do on one side, do exactly the same on the other side.
Isolating variables simplifies the problem and makes equations easier to solve. It helps us visualize the relationship between different quantities.

Mastering Division in Equations

Division is a critical step in the process of isolating a variable. By dividing each side of the equation equally, we can often simplify the equation dramatically.
In our equation \(y = (a+b)x\), we performed division by \((a+b)\) to isolate \(x\). This gives us \(x = \frac{y}{(a+b)}\).
- Division helps to cancel multiplication on the side with the variable. - It reduces the equation to a simpler form where you can easily see what the variable equals.- It's essential to remember never to divide by zero, as division by zero is undefined.
Understanding division in the context of equations helps develop strong problem-solving skills and builds a foundation for tackling more complex mathematical problems.

Simplifying Expressions

Simplifying expressions is the process of reducing complex equations to their simplest form. This involves performing operations like division, as seen in our problem, and reducing fractions or combining like terms.
- The expression \(x = \frac{y}{(a+b)}\) is the result of simplifying the original equation by isolating \(x\). - Simplification makes equations easier to interpret and solve, ensuring clarity of the solution.- Often, simplification can include factoring, distributing, or cancelling out terms.
By simplifying, we make the equation more usable and its solutions more apparent, providing clear paths to solving for unknowns.