Question
What is the solution to the system of equations below?
2 x minus y = 10 and y = negative one-half x + 5
(6, 2)
(6, –2)
(–6, –22)
(–6, 8)
2 x minus y = 10 and y = negative one-half x + 5
(6, 2)
(6, –2)
(–6, –22)
(–6, 8)
Answers
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Answered by
GPT-4o mini
To solve the system of equations given:
1. \( 2x - y = 10 \)
2. \( y = -\frac{1}{2}x + 5 \)
We can use substitution or elimination. Here, we can substitute the second equation into the first equation.
From the second equation, we have:
\[ y = -\frac{1}{2}x + 5 \]
Now substitute this expression for \( y \) into the first equation:
\[
2x - \left(-\frac{1}{2}x + 5\right) = 10
\]
Simplifying this gives:
\[
2x + \frac{1}{2}x - 5 = 10
\]
Combining terms:
\[
2x + \frac{1}{2}x = \frac{4}{2}x + \frac{1}{2}x = \frac{5}{2}x
\]
So we have:
\[
\frac{5}{2}x - 5 = 10
\]
Now, add 5 to both sides:
\[
\frac{5}{2}x = 15
\]
Now, multiply both sides by \(\frac{2}{5}\):
\[
x = 15 \cdot \frac{2}{5} = 6
\]
Next, we substitute \( x = 6 \) back into the second equation to find \( y \):
\[
y = -\frac{1}{2}(6) + 5
\]
\[
y = -3 + 5 = 2
\]
So, the solution to the system of equations is:
\[
(x, y) = (6, 2)
\]
Therefore, the answer is \((6, 2)\).
1. \( 2x - y = 10 \)
2. \( y = -\frac{1}{2}x + 5 \)
We can use substitution or elimination. Here, we can substitute the second equation into the first equation.
From the second equation, we have:
\[ y = -\frac{1}{2}x + 5 \]
Now substitute this expression for \( y \) into the first equation:
\[
2x - \left(-\frac{1}{2}x + 5\right) = 10
\]
Simplifying this gives:
\[
2x + \frac{1}{2}x - 5 = 10
\]
Combining terms:
\[
2x + \frac{1}{2}x = \frac{4}{2}x + \frac{1}{2}x = \frac{5}{2}x
\]
So we have:
\[
\frac{5}{2}x - 5 = 10
\]
Now, add 5 to both sides:
\[
\frac{5}{2}x = 15
\]
Now, multiply both sides by \(\frac{2}{5}\):
\[
x = 15 \cdot \frac{2}{5} = 6
\]
Next, we substitute \( x = 6 \) back into the second equation to find \( y \):
\[
y = -\frac{1}{2}(6) + 5
\]
\[
y = -3 + 5 = 2
\]
So, the solution to the system of equations is:
\[
(x, y) = (6, 2)
\]
Therefore, the answer is \((6, 2)\).
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