Math / Geometry

QuestionFind the equation of the line that is perpendicular to the line $4 x-8 y=7$ and passes through the point $(1,1)$ . $y=-2 x+3$

Studdy Solution

STEP 1

1. The given line's equation is $4x - 8y = 7$ .
2. The line we seek is perpendicular to this given line.
3. The point through which the perpendicular line passes is $(1,1)$ .
4. The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope.
5. The equation of a line can be written in the form $y = mx + c$ , where $m$ is the slope and $c$ is the y-intercept.

STEP 2

1. Determine the slope of the given line.
2. Calculate the slope of the perpendicular line.
3. Use the point-slope form of the equation to find the equation of the perpendicular line.
4. Convert the point-slope form to the slope-intercept form, if necessary, and verify the solution.

STEP 3

Convert the given line's equation $4x - 8y = 7$ to the slope-intercept form $y = mx + c$ to determine its slope.
$4x - 8y = 7$ $-8y = -4x + 7$ $y = \frac{1}{2}x - \frac{7}{8}$
The slope of the given line is $\frac{1}{2}$ .

STEP 4

Calculate the slope of the line that is perpendicular to the given line. The slope of the perpendicular line is the negative reciprocal of $\frac{1}{2}$ .
$m_{\perp} = -\frac{1}{\frac{1}{2}} = -2$

STEP 5

Use the point-slope form of the equation, $y - y_1 = m(x - x_1)$ , with the point $(1,1)$ and the slope $-2$ .
$y - 1 = -2(x - 1)$

STEP 6

Simplify the equation from the point-slope form to the slope-intercept form.
$y - 1 = -2(x - 1)$ $y - 1 = -2x + 2$ $y = -2x + 3$
Therefore, the equation of the line that is perpendicular to $4x - 8y = 7$ and passes through the point $(1,1)$ is:
$y = -2x + 3$

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QuestionFind the equation of the line that is perpendicular to the line $4 x-8 y=7$ and passes through the point $(1,1)$ . $y=-2 x+3$

Studdy Solution

STEP 1

STEP 2

1. Determine the slope of the given line.
2. Calculate the slope of the perpendicular line.
3. Use the point-slope form of the equation to find the equation of the perpendicular line.
4. Convert the point-slope form to the slope-intercept form, if necessary, and verify the solution.

STEP 3

Convert the given line's equation $4x - 8y = 7$ to the slope-intercept form $y = mx + c$ to determine its slope.
$4x - 8y = 7$ $-8y = -4x + 7$ $y = \frac{1}{2}x - \frac{7}{8}$
The slope of the given line is $\frac{1}{2}$ .

STEP 4

Calculate the slope of the line that is perpendicular to the given line. The slope of the perpendicular line is the negative reciprocal of $\frac{1}{2}$ .
$m_{\perp} = -\frac{1}{\frac{1}{2}} = -2$

STEP 5

Use the point-slope form of the equation, $y - y_1 = m(x - x_1)$ , with the point $(1,1)$ and the slope $-2$ .
$y - 1 = -2(x - 1)$

STEP 6

Simplify the equation from the point-slope form to the slope-intercept form.
$y - 1 = -2(x - 1)$ $y - 1 = -2x + 2$ $y = -2x + 3$
Therefore, the equation of the line that is perpendicular to $4x - 8y = 7$ and passes through the point $(1,1)$ is:
$y = -2x + 3$

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QuestionFind the equation of the line that is perpendicular to the line 4x−8y=74 x-8 y=74x−8y=7 and passes through the point (1,1)(1,1)(1,1). y=−2x+3y=-2 x+3y=−2x+3

Studdy Solution

STEP 1

STEP 2

1. Determine the slope of the given line.2. Calculate the slope of the perpendicular line.3. Use the point-slope form of the equation to find the equation of the perpendicular line.4. Convert the point-slope form to the slope-intercept form, if necessary, and verify the solution.

STEP 3

STEP 4

Calculate the slope of the line that is perpendicular to the given line. The slope of the perpendicular line is the negative reciprocal of 12\frac{1}{2}21​.m⊥=−112=−2 m_{\perp} = -\frac{1}{\frac{1}{2}} = -2 m⊥​=−21​1​=−2

STEP 5

Use the point-slope form of the equation, y−y1=m(x−x1)y - y_1 = m(x - x_1)y−y1​=m(x−x1​), with the point (1,1)(1,1)(1,1) and the slope −2-2−2.y−1=−2(x−1) y - 1 = -2(x - 1) y−1=−2(x−1)

STEP 6

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QuestionFind the equation of the line that is perpendicular to the line $4 x-8 y=7$ and passes through the point $(1,1)$ . $y=-2 x+3$

1. Determine the slope of the given line.
2. Calculate the slope of the perpendicular line.
3. Use the point-slope form of the equation to find the equation of the perpendicular line.
4. Convert the point-slope form to the slope-intercept form, if necessary, and verify the solution.

Calculate the slope of the line that is perpendicular to the given line. The slope of the perpendicular line is the negative reciprocal of $\frac{1}{2}$ .
$m_{\perp} = -\frac{1}{\frac{1}{2}} = -2$

Use the point-slope form of the equation, $y - y_1 = m(x - x_1)$ , with the point $(1,1)$ and the slope $-2$ .
$y - 1 = -2(x - 1)$