Question
which method would be best to solve this system
0.5x -1.4y=5.8
y=-0.3x - 1/5
0.5x -1.4y=5.8
y=-0.3x - 1/5
Answers
Answered by
GPT-4o mini
To solve the system of equations
1. \(0.5x - 1.4y = 5.8\)
2. \(y = -0.3x - \frac{1}{5}\)
it might be easiest to use the substitution method because the second equation is already solved for \(y\).
### Steps to Solve Using Substitution:
1. **Substitute the expression for \(y\) from the second equation into the first equation**:
\[
0.5x - 1.4(-0.3x - \frac{1}{5}) = 5.8
\]
2. **Distribute the \(-1.4\)**:
\[
0.5x + 1.4(0.3x) + 1.4(\frac{1}{5}) = 5.8
\]
This simplifies to:
\[
0.5x + 0.42x + \frac{1.4}{5} = 5.8
\]
3. **Combine like terms**:
\[
(0.5 + 0.42)x + \frac{1.4}{5} = 5.8
\]
Which gives:
\[
0.92x + 0.28 = 5.8
\]
4. **Isolate \(x\)**:
\[
0.92x = 5.8 - 0.28
\]
\[
0.92x = 5.52
\]
\[
x = \frac{5.52}{0.92}
\]
\[
x \approx 6
\]
5. **Now, substitute \(x\) back into the second equation to find \(y\)**:
\[
y = -0.3(6) - \frac{1}{5}
\]
\[
y = -1.8 - 0.2 = -2
\]
### Final Solution:
Thus, the solution to the system of equations is:
\[
(x, y) = (6, -2)
\]
This method effectively uses substitution to find the solution. You could also use the elimination method, but in this case, substitution is more straightforward due to the form of the equations.
1. \(0.5x - 1.4y = 5.8\)
2. \(y = -0.3x - \frac{1}{5}\)
it might be easiest to use the substitution method because the second equation is already solved for \(y\).
### Steps to Solve Using Substitution:
1. **Substitute the expression for \(y\) from the second equation into the first equation**:
\[
0.5x - 1.4(-0.3x - \frac{1}{5}) = 5.8
\]
2. **Distribute the \(-1.4\)**:
\[
0.5x + 1.4(0.3x) + 1.4(\frac{1}{5}) = 5.8
\]
This simplifies to:
\[
0.5x + 0.42x + \frac{1.4}{5} = 5.8
\]
3. **Combine like terms**:
\[
(0.5 + 0.42)x + \frac{1.4}{5} = 5.8
\]
Which gives:
\[
0.92x + 0.28 = 5.8
\]
4. **Isolate \(x\)**:
\[
0.92x = 5.8 - 0.28
\]
\[
0.92x = 5.52
\]
\[
x = \frac{5.52}{0.92}
\]
\[
x \approx 6
\]
5. **Now, substitute \(x\) back into the second equation to find \(y\)**:
\[
y = -0.3(6) - \frac{1}{5}
\]
\[
y = -1.8 - 0.2 = -2
\]
### Final Solution:
Thus, the solution to the system of equations is:
\[
(x, y) = (6, -2)
\]
This method effectively uses substitution to find the solution. You could also use the elimination method, but in this case, substitution is more straightforward due to the form of the equations.