Problem 42 - Algebra for College Students

Question

Solve each equation for the indicated variable. \(7(2 x+5 y)=6\) for \(y\)

Step-by-Step Solution

Answer

The solution for \(y\) is \(y = -\frac{2}{5}x + \frac{6}{35}\).

Step 1: Distribute the 7

The given equation is \(7(2x + 5y) = 6\). Start by distributing the 7 across \(2x + 5y\). This yields \(14x + 35y = 6\).

Step 2: Isolate terms with y on one side

The goal is to solve for \(y\), so keep the terms containing \(y\) on one side. To do this, subtract \(14x\) from both sides of the equation: \(35y = 6 - 14x\).

Step 3: Solve for y

Divide every term by 35 to solve for \(y\): \(y = \frac{6 - 14x}{35}\). Simplify the expression to get \(y = \frac{6}{35} - \frac{14x}{35}\). Further simplify to \(y = \frac{6}{35} - \frac{2x}{5}\).

Step 4: Final solution

The equation \(7(2x + 5y) = 6\) solved for \(y\) is \(y = -\frac{2}{5}x + \frac{6}{35}\).

Key Concepts

Variable Isolation

In algebra, variable isolation is the process of solving an equation to express one variable in terms of others. This is a fundamental skill needed for solving and understanding algebraic equations. For example, in the equation \( 14x + 35y = 6 \), if we want to solve for \( y \), the first step is to isolate terms involving \( y \) on one side of the equation. This is done by subtracting \( 14x \) from both sides, resulting in \( 35y = 6 - 14x \). The next step is to solve for \( y \) by ensuring \( y \) is by itself on one side of the equation. We achieve this by dividing every term in the equation by 35, given that we need to isolate \( y \) completely. Understanding this concept is crucial to learn how to handle multiple variables within equations and to solve them one at a time, depending on what the problem is asking for.

Distributive Property

The distributive property is an essential algebraic rule used to simplify expressions. It allows you to multiply a single term by each term inside a parenthesis. In our original exercise, we start with \( 7(2x + 5y) = 6 \). Applying the distributive property, we multiply 7 with both \( 2x \) and \( 5y \), which translates to \( 14x + 35y = 6 \). This property helps in breaking down complex equations into simpler forms, making it easier to isolate variables and find solutions. The resulting expression simplifies the process of getting to the next steps in solving the equation, such as isolating the variables. It is a transferable skill useful across various mathematical problems beyond just algebra.

Equation Simplification

Equation simplification is the process of reducing an equation to its simplest form while preserving its equality. In the given problem, once we isolated the variable \( y \), we have the equation \( y = \frac{6 - 14x}{35} \). Initially, the terms seem complicated, but simplification makes it more readable and easier to interpret.To simplify, we divide both numerator terms by 35, hence breaking it into two parts: \( y = \frac{6}{35} - \frac{14x}{35} \). Further simplification gives \( y = \frac{6}{35} - \frac{2x}{5} \), which makes the equation much easier to comprehend and use. Simplification is beneficial as it highlights the relationship between variables in a straightforward manner, aiding in better understanding and efficient calculation of values.