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Chapter 7: Problem 42

Solve each equation for \(y .\) $$-x+5 y=10$$

Short Answer

Expert verified
\(y = \frac{x + 10}{5}\)

Step by step solution

01

Isolate the Y-Term

The given equation is \(-x + 5y = 10\). First, isolate the term containing \(y\). To do this, add \(x\) to both sides of the equation:\[-x + 5y + x = 10 + x\]Simplify it to:\[5y = x + 10\]
02

Solve for Y

Now, we need to solve for \(y\) by isolating it completely. The current equation is \(5y = x + 10\). To isolate \(y\), divide both sides of the equation by 5:\[y = \frac{x + 10}{5}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving for y
In many algebra problems, solving for a specific variable is key. Here, you need to find the value of \(y\) related to the other terms in the equation. For example, in the equation \(-x + 5y = 10\), your goal is to express \(y\) only in terms of \(x\).
\(\)To start solving for \(y\), identify the terms surrounding \(y\) in the equation. In this case, \(-x\) and \(5\) are affecting \(y\). Understanding this helps you plan the steps to isolate \(y\).
\(\)Approaching problems with the clear objective to solve for \(y\) simplifies many algebra tasks, and architects a clear path through potentially complex equations.
Isolating variables
Isolating variables involves getting your term of interest -- in this case, \(y\) -- alone on one side of the equation. This concept is fundamental in algebra and crucial for rearranging equations.
\(\)In the given equation \(-x + 5y = 10\), the first step is to move any terms that do not include \(y\) to the opposite side of the equation. You add \(x\) to both sides resulting in \(5y = x + 10\).
\(\)Remember:
  • Maintain the balance of the equation by performing the same operation on both sides.
  • Every change should bring you a step closer to having \(y\) alone.
Once terms without \(y\) are cleared from its side, divide by the coefficients next to \(y\). Since \(5\) is multiplied by \(y\), dividing both sides by \(5\) yields:\[y = \frac{x + 10}{5}\]This completes the process of isolating the variable.
Step-by-step solutions
Taking a step-by-step approach to solve equations is one of the best strategies in mathematics. It prevents mistakes and ensures clarity through each stage of problem-solving.
\(\)When you face an equation like \(-x + 5y = 10\), break it down:
  • Step 1: Move terms to isolate the variable you want to solve for. Add \(x\) from the left side to both sides to simplify it.
  • Step 2: Separate \(y\) by dividing both sides by its coefficient, which is \(5\) in this instance.
This step-by-step breakdown transforms what may initially seem complex into simple, manageable tasks. It enhances understanding by allowing you to see clearly how each part of the equation contributes to finding \(y\).
\(\)Ultimately, using a structured approach aids in enhancing your problem-solving abilities significantly. This method fosters a logical flow from transforming the original equation to solving for the desired variable.

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Most popular questions from this chapter

Write an equation in slope-intercept form for each table of values. $$\begin{array}{|r|r|r|r|r|} \hline x & -1 & 0 & 1 & 2 \\ \hline y & -7 & -3 & 1 & 5 \\ \hline \end{array}$$

Graph each equation using the slope and \(y\) -intercept. $$x+y=-3$$

Write an equation in slope-intercept form for the line passing through each pair of points. $$(8,7)and (-9,7)$$

Lorena borrowed \(\$ 100\) from her father and plans to pay him back at a rate of \(\$ 10\) per week. The graph shows the amount Lorena owes her father. Find the equation that represents this relationship. (GRAPH CANNOT COPY) A. \(y=10 x-100\) C. \(y=100-10 x\) B. \(y=10 x-90\) D. \(y=-100-10 x\)

Solve each proportion. $$\frac{5}{10}=\frac{15}{x}$$
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