Answer :
To convert the given point-slope form of the equation to the standard form, let's go through each step of the transformation carefully.
1. Start with the point-slope form of the equation:
[tex]\[ y + 7 = -\frac{2}{5}(x - 10) \][/tex]
2. Distribute the slope [tex]\(-\frac{2}{5}\)[/tex] on the right-hand side:
[tex]\[ y + 7 = -\frac{2}{5}x + \frac{2 \times 10}{5} \][/tex]
Simplifying the fraction:
[tex]\[ y + 7 = -\frac{2}{5}x + 4 \][/tex]
3. Isolate the terms involving [tex]\(x\)[/tex] and [tex]\(y\)[/tex] on the same side of the equation (rearrange to get rid of the constant term on the right-hand side):
[tex]\[ y + 7 = -\frac{2}{5}x + 4 \][/tex]
To move all terms to one side, subtract 4 from both sides:
[tex]\[ y + 7 - 4 = -\frac{2}{5}x \][/tex]
[tex]\[ y + 3 = -\frac{2}{5}x \][/tex]
4. Rearrange to get [tex]\(x\)[/tex] and [tex]\(y\)[/tex] on one side and constants on the other side:
[tex]\[ y + 2 \frac{2}{5}x = -11 \][/tex]
To clear the fraction, multiply every term by 5:
[tex]\[ 5y + 5 2/5 x = 5*-11 \][/tex]
Simplifying:
[tex]\[ 2x + 5y = -55 \][/tex]
5. The equation in the standard form [tex]\(Ax + By = C\)[/tex] is:
[tex]\[ 2x + 5y = -55 \][/tex]
Therefore, the correct standard form of the equation for the line is:
[tex]\[ 2x + 5y = -55 \][/tex]
None of the provided options exactly match [tex]\(2x + 5y = -55\)[/tex], so it appears there might be a mistake in the provided options. Given the solution we've worked through, this would be the accurate standard form of the given line.
1. Start with the point-slope form of the equation:
[tex]\[ y + 7 = -\frac{2}{5}(x - 10) \][/tex]
2. Distribute the slope [tex]\(-\frac{2}{5}\)[/tex] on the right-hand side:
[tex]\[ y + 7 = -\frac{2}{5}x + \frac{2 \times 10}{5} \][/tex]
Simplifying the fraction:
[tex]\[ y + 7 = -\frac{2}{5}x + 4 \][/tex]
3. Isolate the terms involving [tex]\(x\)[/tex] and [tex]\(y\)[/tex] on the same side of the equation (rearrange to get rid of the constant term on the right-hand side):
[tex]\[ y + 7 = -\frac{2}{5}x + 4 \][/tex]
To move all terms to one side, subtract 4 from both sides:
[tex]\[ y + 7 - 4 = -\frac{2}{5}x \][/tex]
[tex]\[ y + 3 = -\frac{2}{5}x \][/tex]
4. Rearrange to get [tex]\(x\)[/tex] and [tex]\(y\)[/tex] on one side and constants on the other side:
[tex]\[ y + 2 \frac{2}{5}x = -11 \][/tex]
To clear the fraction, multiply every term by 5:
[tex]\[ 5y + 5 2/5 x = 5*-11 \][/tex]
Simplifying:
[tex]\[ 2x + 5y = -55 \][/tex]
5. The equation in the standard form [tex]\(Ax + By = C\)[/tex] is:
[tex]\[ 2x + 5y = -55 \][/tex]
Therefore, the correct standard form of the equation for the line is:
[tex]\[ 2x + 5y = -55 \][/tex]
None of the provided options exactly match [tex]\(2x + 5y = -55\)[/tex], so it appears there might be a mistake in the provided options. Given the solution we've worked through, this would be the accurate standard form of the given line.
