You are solving the system of linear equations below. What equation and variable would be best to choose to solve for a variable in terms of the other variable?
[tex]
\begin{array}{l}
2 x-y+7=0 \\
7 x+\frac{1}{2} x=7
\end{array}
[/tex]
A. Solve for [tex]$x$[/tex] in the second equation.
B. Solve for [tex]$y$[/tex] in the second equation.
C. Solve for [tex]$y$[/tex] in the first equation.
D. Solve for [tex]$x$[/tex] in the first equation.
Answer (2)
x = 15 14 , y = 2 x + 7 , x = 2 y − 7
Explanation
Understanding the Problem We are given a system of two linear equations with two variables, x and y . Our goal is to solve for x in the second equation, solve for y in the second equation, solve for y in the first equation, and solve for x in the first equation.
Solving for x in the Second Equation Let's start with the second equation: 7 x + 2 1 x = 7 . We can combine the terms with x to get 2 15 x = 7 . To solve for x , we multiply both sides of the equation by 15 2 : x = 7 × 15 2 = 15 14 So, x = 15 14 .
Solving for y in the Second Equation Now, let's try to solve for y in the second equation. Notice that the second equation, 7 x + 2 1 x = 7 , does not contain the variable y . Therefore, it's not possible to solve for y using this equation. The value of y can be anything.
Solving for y in the First Equation Next, let's solve for y in the first equation: 2 x − y + 7 = 0 . To isolate y , we can add y to both sides of the equation: 2 x + 7 = y . Thus, y = 2 x + 7 .
Solving for x in the First Equation Finally, let's solve for x in the first equation: 2 x − y + 7 = 0 . To isolate x , we first subtract 7 and add y to both sides: 2 x = y − 7 . Then, we divide both sides by 2: x = 2 y − 7 .
Final Answers In summary:
Solving for x in the second equation gives x = 15 14 .
It is not possible to solve for y in the second equation.
Solving for y in the first equation gives y = 2 x + 7 .
Solving for x in the first equation gives x = 2 y − 7 .
Examples
Understanding how to manipulate equations to solve for specific variables is a fundamental skill in algebra and has numerous real-world applications. For example, if you're planning a road trip and need to determine how long it will take to reach your destination, you can use the equation distance = speed × time. If you know the distance and your average speed, you can rearrange the equation to solve for time: time = distance / speed. This allows you to calculate the estimated travel time. Similarly, in budgeting, you can rearrange equations to solve for different financial variables, such as calculating the monthly payment needed to pay off a loan within a specific timeframe.
The best choice is to solve for x in the second equation. Solving gives x = 15 14 . Other options either lead to no solution or are less straightforward.
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