Question
Consider the following system of equations: 13x + y = 3 2y = −23x + 10 Solve the system of equations using the elimination method. Be sure to show all of your work to earn full points. (5 points)
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To solve the given system of equations using the elimination method, we start with the two equations:
1) \( 13x + y = 3 \)
2) \( 2y = -23x + 10 \)
First, let's rearrange equation (2) to solve for \(y\):
\[ 2y = -23x + 10 \]
To isolate \(y\), divide the entire equation by 2:
\[ y = -\frac{23}{2}x + 5 \]
Now we have:
1) \( 13x + y = 3 \)
2) \( y = -\frac{23}{2}x + 5 \)
Next, we will substitute equation (2) into equation (1) to eliminate \(y\):
Substituting for \(y\) in equation (1):
\[ 13x + \left(-\frac{23}{2}x + 5\right) = 3 \]
Now, simplify the equation:
\[ 13x - \frac{23}{2}x + 5 = 3 \]
To combine the terms involving \( x \), let's convert \( 13x \) into a fraction:
\[ 13x = \frac{26}{2}x \]
Now the equation becomes:
\[ \frac{26}{2}x - \frac{23}{2}x + 5 = 3 \]
Combine the \( x \) terms:
\[ \frac{26 - 23}{2} x + 5 = 3 \]
Simplifying that gives:
\[ \frac{3}{2} x + 5 = 3 \]
Next, isolate \( \frac{3}{2} x \):
\[ \frac{3}{2} x = 3 - 5 \]
\[ \frac{3}{2} x = -2 \]
To solve for \( x \), multiply both sides by the reciprocal of \( \frac{3}{2} \):
\[ x = -2 \cdot \frac{2}{3} \]
\[ x = -\frac{4}{3} \]
Now that we have \( x \), we substitute it back into one of the original equations to find \( y \). Using equation (1):
\[ 13\left(-\frac{4}{3}\right) + y = 3 \]
Calculating \( 13(-\frac{4}{3}) \):
\[ -\frac{52}{3} + y = 3 \]
Next, convert \(3\) into a fraction with a denominator of 3:
\[ 3 = \frac{9}{3} \]
Now the equation looks like this:
\[ -\frac{52}{3} + y = \frac{9}{3} \]
Isolating \(y\):
\[ y = \frac{9}{3} + \frac{52}{3} \]
\[ y = \frac{9 + 52}{3} \]
\[ y = \frac{61}{3} \]
Thus, the solution to the system of equations is:
\[ x = -\frac{4}{3} \]
\[ y = \frac{61}{3} \]
We can summarize the solution as:
**Solution:**
\[ \left( x, y \right) = \left( -\frac{4}{3}, \frac{61}{3} \right) \]
1) \( 13x + y = 3 \)
2) \( 2y = -23x + 10 \)
First, let's rearrange equation (2) to solve for \(y\):
\[ 2y = -23x + 10 \]
To isolate \(y\), divide the entire equation by 2:
\[ y = -\frac{23}{2}x + 5 \]
Now we have:
1) \( 13x + y = 3 \)
2) \( y = -\frac{23}{2}x + 5 \)
Next, we will substitute equation (2) into equation (1) to eliminate \(y\):
Substituting for \(y\) in equation (1):
\[ 13x + \left(-\frac{23}{2}x + 5\right) = 3 \]
Now, simplify the equation:
\[ 13x - \frac{23}{2}x + 5 = 3 \]
To combine the terms involving \( x \), let's convert \( 13x \) into a fraction:
\[ 13x = \frac{26}{2}x \]
Now the equation becomes:
\[ \frac{26}{2}x - \frac{23}{2}x + 5 = 3 \]
Combine the \( x \) terms:
\[ \frac{26 - 23}{2} x + 5 = 3 \]
Simplifying that gives:
\[ \frac{3}{2} x + 5 = 3 \]
Next, isolate \( \frac{3}{2} x \):
\[ \frac{3}{2} x = 3 - 5 \]
\[ \frac{3}{2} x = -2 \]
To solve for \( x \), multiply both sides by the reciprocal of \( \frac{3}{2} \):
\[ x = -2 \cdot \frac{2}{3} \]
\[ x = -\frac{4}{3} \]
Now that we have \( x \), we substitute it back into one of the original equations to find \( y \). Using equation (1):
\[ 13\left(-\frac{4}{3}\right) + y = 3 \]
Calculating \( 13(-\frac{4}{3}) \):
\[ -\frac{52}{3} + y = 3 \]
Next, convert \(3\) into a fraction with a denominator of 3:
\[ 3 = \frac{9}{3} \]
Now the equation looks like this:
\[ -\frac{52}{3} + y = \frac{9}{3} \]
Isolating \(y\):
\[ y = \frac{9}{3} + \frac{52}{3} \]
\[ y = \frac{9 + 52}{3} \]
\[ y = \frac{61}{3} \]
Thus, the solution to the system of equations is:
\[ x = -\frac{4}{3} \]
\[ y = \frac{61}{3} \]
We can summarize the solution as:
**Solution:**
\[ \left( x, y \right) = \left( -\frac{4}{3}, \frac{61}{3} \right) \]
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.