GRE Quantitative Reasoning - Coordinate Geometry

Find the distance from point $\left ( 1,-\frac{1}{2}\right )$ to the line .

$\frac{11}{2}$

$\frac{9}{2}$

$-\frac{11}{2}$

$-\frac{9}{2}$

Explanation

Draw a line that connects the point and intersects the line at a perpendicular angle.

The vertical distance from the point $\left ( 1,-\frac{1}{2}\right )$ to the line will be the difference of the 2 y-values.

The distance can never be negative.

$5-\left(-\frac{1}{2}\right) = \frac{10}{2}+\frac{1}{2}=\frac{11}{2}$

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:

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

Find the distance from point $\left ( 1,-\frac{1}{2}\right )$ to the line .

$\frac{11}{2}$

$\frac{9}{2}$

$-\frac{11}{2}$

$-\frac{9}{2}$

Explanation

Draw a line that connects the point and intersects the line at a perpendicular angle.

The vertical distance from the point $\left ( 1,-\frac{1}{2}\right )$ to the line will be the difference of the 2 y-values.

The distance can never be negative.

$5-\left(-\frac{1}{2}\right) = \frac{10}{2}+\frac{1}{2}=\frac{11}{2}$

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

Find the slope of a line that passes through and

$m=\frac{1}{2}$

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

Explanation

The formula for slope is:

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

$m = \frac{5-(-1)}{6-3}$

$m = \frac{5+1}{6-3}$

$m = \frac{6}{3}$

What is the slope and y-intercept of this line: ?

$Slope=\frac {3}{4},y-intercept=3$

$Slope=\frac {3}{4},y-intercept=2$

$Slope=\frac {-3}{4},y-intercept=3$

$Slope=\frac {4}{3},y-intercept=3$

Explanation

Step 1: Move the y-term to the other side. We will add 4y.

$\rightarrow 3x=12+4y$

Step 2: Move the constant to the other side. We will subtract 12.

$\rightarrow 3x-12=4y$

Step 3: Divide by the coefficient in front of the y term. In this question, we divide all terms by 4.

$\frac {3x-12}{4}=\frac {4y}{4}\rightarrow\frac {3}{4}x+\frac {-12}{4}=y$

$\rightarrow y=\frac {3}{4}x-3$

Step 4: Identify the slope and the y-intercept. The slope is the number that is in front of the x term, and the y-intercept is the number that comes after the x-term.

In this question, the slope is $\frac {3}{4}$ and the y-intercept is .

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:

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

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

Find the slope of a line that passes through and

$m=\frac{1}{2}$

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

Explanation

The formula for slope is:

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

$m = \frac{5-(-1)}{6-3}$

$m = \frac{5+1}{6-3}$

$m = \frac{6}{3}$

What is the slope and y-intercept of this line: ?

$Slope=\frac {3}{4},y-intercept=3$

$Slope=\frac {3}{4},y-intercept=2$

$Slope=\frac {-3}{4},y-intercept=3$

$Slope=\frac {4}{3},y-intercept=3$

Explanation

Step 1: Move the y-term to the other side. We will add 4y.

$\rightarrow 3x=12+4y$

Step 2: Move the constant to the other side. We will subtract 12.

$\rightarrow 3x-12=4y$

Step 3: Divide by the coefficient in front of the y term. In this question, we divide all terms by 4.

$\frac {3x-12}{4}=\frac {4y}{4}\rightarrow\frac {3}{4}x+\frac {-12}{4}=y$

$\rightarrow y=\frac {3}{4}x-3$

Step 4: Identify the slope and the y-intercept. The slope is the number that is in front of the x term, and the y-intercept is the number that comes after the x-term.

In this question, the slope is $\frac {3}{4}$ and the y-intercept is .

Coordinate Geometry

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