Find the best answers to your questions at Westonci.ca, where experts and enthusiasts provide accurate, reliable information. Discover comprehensive solutions to your questions from a wide network of experts on our user-friendly platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
To find the equation of the line that passes through the points [tex]\((5, -2)\)[/tex] and [tex]\((-3, 4)\)[/tex], let's go through the process step by step:
1. Compute the slope [tex]\(m\)[/tex] of the line:
The slope [tex]\(m\)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the given points [tex]\((x_1, y_1) = (5, -2)\)[/tex] and [tex]\((x_2, y_2) = (-3, 4)\)[/tex]:
[tex]\[ m = \frac{4 - (-2)}{-3 - 5} = \frac{4 + 2}{-3 - 5} = \frac{6}{-8} = -\frac{3}{4} \][/tex]
2. Find the y-intercept [tex]\(b\)[/tex]:
We use the point-slope form of the linear equation [tex]\(y = mx + b\)[/tex] and one of the points to find the y-intercept [tex]\(b\)[/tex]. Using the point [tex]\((5, -2)\)[/tex] as an example:
[tex]\[ -2 = -\frac{3}{4} \cdot 5 + b \][/tex]
Simplifying and solving for [tex]\(b\)[/tex]:
[tex]\[ -2 = -\frac{15}{4} + b \\ -2 + \frac{15}{4} = b \\ -\frac{8}{4} + \frac{15}{4} = b \\ \frac{7}{4} = b \][/tex]
3. Write the equation in slope-intercept form:
Now we have the slope [tex]\(m = -\frac{3}{4}\)[/tex] and the y-intercept [tex]\(b = \frac{7}{4}\)[/tex]. The slope-intercept form of the line is:
[tex]\[ y = -\frac{3}{4}x + \frac{7}{4} \][/tex]
4. Convert to standard form [tex]\(Ax + By + C = 0\)[/tex]:
Multiplying the entire equation by 4 to clear the fractions:
[tex]\[ 4y = -3x + 7 \][/tex]
Rearrange to standard form:
[tex]\[ 3x + 4y - 7 = 0 \][/tex]
Therefore, the equation of the line passing through the points [tex]\((5, -2)\)[/tex] and [tex]\((-3, 4)\)[/tex] is
[tex]\[ 3x + 4y - 7 = 0 \][/tex]
So, the correct answer is:
[tex]\[ 3 x + 4 y - 7 = 0 \][/tex]
1. Compute the slope [tex]\(m\)[/tex] of the line:
The slope [tex]\(m\)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the given points [tex]\((x_1, y_1) = (5, -2)\)[/tex] and [tex]\((x_2, y_2) = (-3, 4)\)[/tex]:
[tex]\[ m = \frac{4 - (-2)}{-3 - 5} = \frac{4 + 2}{-3 - 5} = \frac{6}{-8} = -\frac{3}{4} \][/tex]
2. Find the y-intercept [tex]\(b\)[/tex]:
We use the point-slope form of the linear equation [tex]\(y = mx + b\)[/tex] and one of the points to find the y-intercept [tex]\(b\)[/tex]. Using the point [tex]\((5, -2)\)[/tex] as an example:
[tex]\[ -2 = -\frac{3}{4} \cdot 5 + b \][/tex]
Simplifying and solving for [tex]\(b\)[/tex]:
[tex]\[ -2 = -\frac{15}{4} + b \\ -2 + \frac{15}{4} = b \\ -\frac{8}{4} + \frac{15}{4} = b \\ \frac{7}{4} = b \][/tex]
3. Write the equation in slope-intercept form:
Now we have the slope [tex]\(m = -\frac{3}{4}\)[/tex] and the y-intercept [tex]\(b = \frac{7}{4}\)[/tex]. The slope-intercept form of the line is:
[tex]\[ y = -\frac{3}{4}x + \frac{7}{4} \][/tex]
4. Convert to standard form [tex]\(Ax + By + C = 0\)[/tex]:
Multiplying the entire equation by 4 to clear the fractions:
[tex]\[ 4y = -3x + 7 \][/tex]
Rearrange to standard form:
[tex]\[ 3x + 4y - 7 = 0 \][/tex]
Therefore, the equation of the line passing through the points [tex]\((5, -2)\)[/tex] and [tex]\((-3, 4)\)[/tex] is
[tex]\[ 3x + 4y - 7 = 0 \][/tex]
So, the correct answer is:
[tex]\[ 3 x + 4 y - 7 = 0 \][/tex]
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.