Problem 66 - Algebra 2

Question

Solve each equation or formula for the specified variable. $$ 3 x+7 x y=-2, \text { for } x $$

Step-by-Step Solution

Answer

\(x = \frac{-2}{3 + 7y}\)

Step 1: Combine like terms

Start by simplifying the equation \(3x + 7xy = -2\). Notice that both terms on the left side have \(x\) as a factor. Rewrite the equation as \(x(3 + 7y) = -2\) by factoring out \(x\).

Step 2: Isolate \(x\)

To solve for \(x\), divide both sides of the equation by \((3 + 7y)\). The equation then becomes \(x = \frac{-2}{3 + 7y}\). This isolates \(x\) on the left side of the equation.

Key Concepts

Combining Like Terms

When solving equations, the first important step is often to combine like terms. This means you group similar terms in an equation to simplify it for easier solving. In our exercise, the equation is given as \(3x + 7xy = -2\). Here, observe that the terms \(3x\) and \(7xy\) both contain the variable \(x\).

• Identify terms that can be combined by determining if they have common variables or coefficients.
• In this case, both terms share \(x\) as a common factor.

After grouping these terms, rewriting them with a common factor is the subsequent step. By factoring \(x\) as a common factor, the expression becomes \(x(3 + 7y) = -2\). This simplification is crucial because it makes dealing with the equation more straightforward in the next steps.

Factoring

Factoring is a powerful technique that simplifies expressions, making it easier to solve for variables. In our context, factoring involves pulling out a common factor from the terms in the equation. For instance, when we transform \(3x + 7xy = -2\) into \(x(3 + 7y) = -2\), we are taking the common factor \(x\) out of the equation.

• Identify common factors in the equation's terms.
• Future equations can often be solved effortlessly once factors are isolated.

Factoring is crucial because it consolidates terms, reducing the complexity of expressions and laying the groundwork for isolating variables.

Isolation of Variables

Isolating variables is often the final goal in solving equations, as it allows us to express a specific variable explicitly. With the equation \(x(3 + 7y) = -2\), we are focused on isolating \(x\). To do this, divide both sides of the equation by the term \((3 + 7y)\).

• The division step effectively "moves" the unwanted factor to the other side of the equation.
• This is the mathematical manipulation that isolates \(x\) alone on one side.

Thus, after dividing, we derive that \(x = \frac{-2}{3 + 7y}\). This equation clearly shows \(x\) on its own, indicating it has been isolated successfully. This step is crucial for deducing explicit form solutions to equations involving multiple variables.