Finding Equations of Lines

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GRE Quantitative Reasoning › Finding Equations of Lines

Questions 1 - 7

1

What is the equation of a line that passes through points and in slope-intercept form?

Explanation

To find the equation of the line, first find the slope using the formula:

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

The points that the line passes through are and .

$m = \frac{-2-4}{1-3}$

$m= \frac{-6}{-2}$

Then pick one set of points and place in the form . Either set of points will give you the same equation. Points were used.

$4 = 3\times 3 + b$

Subtract from both sides of the equation.

The equation of the line in slope-intercept form or is

2

What is the equation of the line (in slope-intercept form) that goes through the points: and ?

$y=\frac {4}{5}x+\frac {12}{5}$

$y=\frac {4}{5}x-\frac {12}{5}$

$y=-\frac {4}{5}x+\frac {12}{5}$

$y=-\frac {4}{5}x-\frac {12}{5}$

Explanation

Step 1: Find the slope between the two points:

$\frac {y_2-y_1}{x_2-x_1}=\frac {8-4}{7-2}=\frac {4}{5}$

Step 2: Write the slope-intercept form:

$y=\frac {4}{5}x+b$

Step 3. Find b. Plug in (x,y) from one of the points:

$4=\frac {4}{5}(2)+b$

$4=\frac {8}{5}+b$

$4-\frac {8}{5}=b$

$\frac {20-8}{5}=b$

$\frac {12}{5}=b$
Step 4: Write out the full equation:

$y=\frac {4}{5}x+\frac {12}{5}$

3

Find the equation of a line in slope-intercept form that passes through points and has a slope of

Explanation

To find the equation of a line given points and use the point- slope formula:

$y-y_{1} = m(x -x_{1})$

Distribute the to what is inside the parenthesis.

$-2\times 5 = -10$

Subtract from both sides of the equation.

When the sign is the same for both integers, add.

The equation of the line in slope-intercept form is:

4

Find the equation of a line in slope-intercept form that passes though points and .

Explanation

To find the equation of a line that passes through and , first find the slope using this formula:

$m = \frac{y_{2} -y_{1}}{x_{2}-x_{1}}$

$m = \frac{5-3}{-2-(-1)} = \frac{2}{-1}$

Using one set of points or coordinates and the value of slope, plug these values into:

Either set of points will give you the same equation of the line.

Coordinates .

Distribute the to what is inside the parenthesis.

$-2\times x = -2x$

$-2 \times 1 = -2$

Add to both sides of the equation.

The equation of the line in slope-intercept form is:

5

Find the equation of the line with points

Explanation

To solve for you must use the equation $(y^{1}-y^{2})/(x^{1}-x^{2})$

To solve for be we must plug in one of the points

simplify

Add to both sides

6

A line has a slope of $m=-\frac{1}{4},$ and goes through point What is the equation of the line in slope-intercept form?

$y = -\frac{1}{4} +8$

$y = -\frac{1}{4} -8$

$y = \frac{1}{4} +8$

$y = \frac{1}{4} - 8$

Explanation

The slope-intercept form is , where represents the slope, and represent the points, and is the y-intercept or the value of when

The slope or $\frac{rise}{run}$ has been given as $m=-\frac{1}{4}$ .

The points that this equation of the line passes through are and

because based on the points, when That is the y-intercept.

The equation of the line in slope-intercept form is:

$y=-\frac{1}{4} x+8$

7

Find a line through the point perpendicular to the line .

Explanation

To solve this problem you must first find the slope of the original equation by point it into y-mx+b form

subtract x from both sides

Then you must find the reciprocal of the slope to get the slope of the perpendicular line

reciprocal is

Finally you must you point-slope form to solve

$y-y_{1}=m(x-x_{1})$

Multiply through the parentheses

add three to both sides

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