Common Core: 8th Grade Math : Expressions & Equations

Study concepts, example questions & explanations for Common Core: 8th Grade Math

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Example Questions

Example Question #1 : Use Similar Triangles To Show Equal Slopes: Ccss.Math.Content.8.Ee.B.6

The equation of a line is \(\displaystyle 12x-8y=6\). Find the slope of this line.

Possible Answers:

\(\displaystyle -\frac{3}{2}\)

\(\displaystyle -\frac{3}{4}\)

\(\displaystyle \frac{3}{2}\)

\(\displaystyle \frac{3}{4}\)

Correct answer:

\(\displaystyle \frac{3}{2}\)

Explanation:

To find the slope, you will need to put the equation in \(\displaystyle y=mx+b\) form. The value of \(\displaystyle m\) will be the slope.

\(\displaystyle 12x-8y=6\)

Subtract \(\displaystyle 6\) from either side:

\(\displaystyle 8y=12x-6\)

Divide each side by \(\displaystyle 8\):

\(\displaystyle y=\frac{3}{2}x-\frac{3}{4}\)

You can now easily identify the value of \(\displaystyle m\).

\(\displaystyle m=\frac{3}{2}\)

Example Question #1 : How To Find X Or Y Intercept

What is the \(\displaystyle y\)-intercept of the graph of the function \(\displaystyle f \left ( x\right ) = 2x^{2} - 7x + 5\) ?

Possible Answers:

\(\displaystyle \left (0,-7 \right )\)

\(\displaystyle \left (0,2 \right )\)

\(\displaystyle \left (0,0 \right )\)

\(\displaystyle \left (0,5 \right )\)

\(\displaystyle \left ( 0, -3 \frac{1}{2}\right )\)

Correct answer:

\(\displaystyle \left (0,5 \right )\)

Explanation:

The \(\displaystyle y\)-intercept of the graph of a function is the point at which it intersects the \(\displaystyle y\)-axis - that is, at which \(\displaystyle x = 0\). This point is \(\displaystyle \left (0, f(0) \right )\), so evaluate \(\displaystyle f(0)\):

\(\displaystyle f \left ( x\right ) = 2x^{2} - 7x + 5\)

\(\displaystyle f \left (0\right ) = 2 \cdot 0^{2} - 7 \cdot 0 + 5\)

\(\displaystyle f \left (0\right ) = 0- 0 + 5 = 5\)

The \(\displaystyle y\)-intercept is \(\displaystyle \left (0,5 \right )\).

Example Question #2 : Use Similar Triangles To Show Equal Slopes: Ccss.Math.Content.8.Ee.B.6

Give the \(\displaystyle y\)-intercept, if there is one, of the graph of the equation

\(\displaystyle y = \frac{1}{x^{2}+ x} + 3\)

Possible Answers:

\(\displaystyle (0,4)\)

The graph has no \(\displaystyle y\)-intercept.

\(\displaystyle \left ( 0, \frac{3}{2}\right )\)

\(\displaystyle (0,2)\)

\(\displaystyle (0,3)\)

Correct answer:

The graph has no \(\displaystyle y\)-intercept.

Explanation:

The \(\displaystyle y\)-intercept is the point at which the graph crosses the \(\displaystyle y\)-axis; at this point, the \(\displaystyle x\)-coordinate is 0, so substitute \(\displaystyle 0\) for \(\displaystyle x\) in the equation:

\(\displaystyle y = \frac{1}{x^{2}+ x} + 3\)

\(\displaystyle y = \frac{1}{0^{2}+0} + 3\)

\(\displaystyle y = \frac{1}{0} + 3\)

However, since this expression has 0 in a denominator, it is of undefined value. This means that there is no value of \(\displaystyle x\) paired with \(\displaystyle y\)-coordinate 0, and, subsequently, the graph of the equation has no \(\displaystyle y\)-intercept.

Example Question #2 : How To Find X Or Y Intercept

Give the \(\displaystyle y\)-intercept, if there is one, of the graph of the equation

\(\displaystyle y = \frac{3}{\left | 3x-8\right |}\)

Possible Answers:

\(\displaystyle \left ( 0,- \frac{3}{5}\right )\)

\(\displaystyle \left ( 0, \frac{3}{5}\right )\)

\(\displaystyle \left ( 0, -\frac{3}{8}\right )\)

The graph has no \(\displaystyle y\)-intercept.

\(\displaystyle \left ( 0, \frac{3}{8}\right )\)

Correct answer:

\(\displaystyle \left ( 0, \frac{3}{8}\right )\)

Explanation:

The \(\displaystyle y\)-intercept is the point at which the graph crosses the \(\displaystyle y\)-axis; at this point, the \(\displaystyle x\)-coordinate is 0, so substitute \(\displaystyle 0\) for \(\displaystyle x\) in the equation:

\(\displaystyle y = \frac{3}{\left | 3x-8\right |}\)

\(\displaystyle y = \frac{3}{\left | 3 \cdot 0 -8\right |}\)

\(\displaystyle y = \frac{3}{\left | 0 -8\right |}\)

\(\displaystyle y = \frac{3}{\left | -8\right |}\)

\(\displaystyle y = \frac{3}{ 8}\)

The \(\displaystyle y\)-intercept is \(\displaystyle \left ( 0, \frac{3}{8}\right )\).

Example Question #3 : Use Similar Triangles To Show Equal Slopes: Ccss.Math.Content.8.Ee.B.6

Give the \(\displaystyle y\)-intercept, if there is one, of the graph of the equation

\(\displaystyle y = 3(x-2)^{2} + 4 (x-7)\).

Possible Answers:

\(\displaystyle (0, -16)\)

\(\displaystyle (0, 8)\)

The graph does not have a \(\displaystyle y\)-intercept.

\(\displaystyle (0, -42)\)

\(\displaystyle (0, -64)\)

Correct answer:

\(\displaystyle (0, -16)\)

Explanation:

The \(\displaystyle y\)-intercept is the point at which the graph crosses the \(\displaystyle y\)-axis; at this point, the \(\displaystyle x\)-coordinate is 0, so substitute \(\displaystyle 0\) for \(\displaystyle x\) in the equation:

\(\displaystyle y = 3(x-2)^{2} + 4 (x-7)\)

\(\displaystyle y = 3(0-2)^{2} + 4 (0-7)\)

\(\displaystyle y = 3( -2)^{2} + 4 ( -7)\)

\(\displaystyle y = 3 \cdot 4 + 4 ( -7)\)

\(\displaystyle y = 12+ 4 ( -7)\)

\(\displaystyle y = 12+ (-28)\)

\(\displaystyle y = -16\)

The \(\displaystyle y\)-intercept is the point \(\displaystyle (0, -16)\)

Example Question #4 : Use Similar Triangles To Show Equal Slopes: Ccss.Math.Content.8.Ee.B.6

A line passes through \(\displaystyle (5, 2)\) and is perpendicular to the line of the equation \(\displaystyle 3x-4y = 17\). Give the \(\displaystyle x\)-intercept of this line.

Possible Answers:

The line has no \(\displaystyle x\)-intercept.

\(\displaystyle \left (2 \frac{1}{3}, 0 \right )\)

\(\displaystyle \left ( -1\frac{3}{4}, 0\right )\)

\(\displaystyle \left ( 8\frac{2}{3}, 0\right )\)

\(\displaystyle \left (6 \frac{1}{2}, 0 \right )\)

Correct answer:

\(\displaystyle \left (6 \frac{1}{2}, 0 \right )\)

Explanation:

First, find the slope of the second line \(\displaystyle 3x-4y = 17\) by solving for \(\displaystyle y\) as follows:

\(\displaystyle 3x-4y = 17\)

\(\displaystyle 3x-4y +4y - 17 = 17+4y - 17\)

\(\displaystyle 4y = 3x-17\)

\(\displaystyle \frac{4y }{4}=\frac{ 3x-17 }{4}\)

\(\displaystyle y = \frac{3}{4}x- \frac{17}{4}\)

The equation is now in the slope-intercept form \(\displaystyle y = mx+ b\); the slope of the second line is the \(\displaystyle x\)-coefficient \(\displaystyle m= \frac{3}{4}\).

The first line, being perpendicular to the second, has as its slope the opposite of the reciprocal of \(\displaystyle \frac{3}{4}\), which is \(\displaystyle m=- \frac{4}{3}\).

Therefore, we are looking for a line through \(\displaystyle (5, 2)\) with slope \(\displaystyle m=- \frac{4}{3}\). Using point-slope form

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

with 

\(\displaystyle x _{1 } =5, y _{1 } = 2,m=- \frac{4}{3}\),

the equation becomes

\(\displaystyle y - 2 =- \frac{4}{3}(x -5)\).

To find the \(\displaystyle x\)-intercept, substitute 0 for \(\displaystyle y\) and solve for \(\displaystyle x\):

\(\displaystyle 0 - 2 =- \frac{4}{3}(x -5)\)

\(\displaystyle - 2 =- \frac{4}{3}x+ \frac{20}{3}\)

\(\displaystyle - 2- \frac{20}{3} =- \frac{4}{3}x+ \frac{20}{3} - \frac{20}{3}\)

\(\displaystyle - \frac{26}{3} =- \frac{4}{3}x\)

\(\displaystyle - \frac{3} {4}\cdot\left (- \frac{26}{3} \right )=- \frac{3} {4}\cdot \left (- \frac{4}{3}x \right )\)

\(\displaystyle x= \frac{26}{4} = \frac{13}{2} = 6\frac{1}{2}\)

The  \(\displaystyle x\)-intercept is the point \(\displaystyle \left (6 \frac{1}{2}, 0 \right )\).

Example Question #273 : Geometry

A line passes through \(\displaystyle (5, 2)\) and is parallel to the line of the equation \(\displaystyle 3x-4y = 17\). Give the \(\displaystyle x\)-intercept of this line.

Possible Answers:

\(\displaystyle \left ( 8\frac{2}{3}, 0\right )\)

The line has no \(\displaystyle x\)-intercept.

\(\displaystyle \left (2 \frac{1}{3}, 0 \right )\)

\(\displaystyle \left ( -1\frac{3}{4}, 0\right )\)

\(\displaystyle \left (6 \frac{1}{2}, 0 \right )\)

Correct answer:

\(\displaystyle \left (2 \frac{1}{3}, 0 \right )\)

Explanation:

First, find the slope of the second line \(\displaystyle 3x-4y = 17\) by solving for \(\displaystyle y\) as follows:

\(\displaystyle 3x-4y = 17\)

\(\displaystyle 3x-4y +4y - 17 = 17+4y - 17\)

\(\displaystyle 4y = 3x-17\)

\(\displaystyle \frac{4y }{4}=\frac{ 3x-17 }{4}\)

\(\displaystyle y = \frac{3}{4}x- \frac{17}{4}\)

The equation is now in the slope-intercept form \(\displaystyle y = mx+ b\); the slope of the second line is the \(\displaystyle x\)-coefficient \(\displaystyle m= \frac{3}{4}\).

The first line, being parallel to the second, has the same slope. 

Therefore, we are looking for a line through \(\displaystyle (5, 2)\) with slope \(\displaystyle m= \frac{3}{4}\). Using point-slope form

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

with 

\(\displaystyle x _{1 } =5, y _{1 } = 2,m= \frac{3}{4}\),

the equation becomes

\(\displaystyle y - 2 = \frac{3}{4}(x -5)\).

To find the \(\displaystyle x\)-intercept, substitute 0 for \(\displaystyle y\) and solve for \(\displaystyle x\):

\(\displaystyle 0 - 2 = \frac{3}{4}(x -5)\)

\(\displaystyle - 2 = \frac{3}{4} x - \frac{15}{4}\)

\(\displaystyle - 2 + \frac{15}{4}= \frac{3}{4} x - \frac{15}{4} + \frac{15}{4}\)

\(\displaystyle \frac{3}{4} x = \frac{7}{4}\)

\(\displaystyle \frac{4} {3}\cdot \frac{3}{4} x =\frac{4} {3}\cdot \frac{7}{4}\)

\(\displaystyle x = \frac{7}{3} = 2\frac{1}{3}\)

The \(\displaystyle x\)-intercept is the point \(\displaystyle \left (2 \frac{1}{3}, 0 \right )\).

Example Question #274 : Geometry

Give the \(\displaystyle y\)-intercept of the line with slope \(\displaystyle \frac{2}{5}\) that passes through point \(\displaystyle (5, 9)\).

Possible Answers:

\(\displaystyle \left (0, -27 \frac{1}{2} \right )\)

\(\displaystyle (0, 7)\)

\(\displaystyle \left (0, -17 \frac{1}{2} \right )\)

The line has no \(\displaystyle y\)-intercept.

\(\displaystyle (0, -11)\)

Correct answer:

\(\displaystyle (0, 7)\)

Explanation:

By the point-slope formula, this line has the equation

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

where

\(\displaystyle x_{1} = 5, y_{1} = 9 ,m = \frac{2}{5}\)

By substitution, the equation becomes

\(\displaystyle y -9 = \frac{2}{5}(x -5)\) 

To find the \(\displaystyle y\)-intercept, substitute 0 for \(\displaystyle x\) and solve for \(\displaystyle y\):

\(\displaystyle y -9 = \frac{2}{5}(0 -5)\)

\(\displaystyle y -9 = \frac{2}{5}( -5)\)

\(\displaystyle y - 9 = -2\)

\(\displaystyle y = 7\)

The \(\displaystyle y\)-intercept is the point \(\displaystyle (0, 7)\).

Example Question #275 : Geometry

Give the \(\displaystyle y\)-intercept of the line that passes through points \(\displaystyle (-3, 4)\) and \(\displaystyle (2, -3)\).

Possible Answers:

\(\displaystyle \left (0, - 5\frac{6}{7} \right )\)

\(\displaystyle \left (0, - \frac{1}{5} \right )\)

\(\displaystyle \left (0, - \frac{1}{7} \right )\)

The line has no \(\displaystyle y\)-intercept.

\(\displaystyle \left (0, - 8 \frac{1}{5} \right )\)

Correct answer:

\(\displaystyle \left (0, - \frac{1}{5} \right )\)

Explanation:

First, find the slope of the line, using the slope formula

\(\displaystyle m = \frac{y_{2} -y_{1}}{x_{2}-x_{1}}\)

setting \(\displaystyle x_{1} = -3, y_{1} = 4, x_{2} = 2, y_{2} = -3\):

\(\displaystyle m = \frac{-3 -4}{2-(-3)} = \frac{-7}{5} =- \frac{7}{5}\)

By the point-slope formula, this line has the equation

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

where

\(\displaystyle x_{1} = -3, y_{1} = 4, m = - \frac{7}{5}\); the line becomes

\(\displaystyle y - 4 = - \frac{7}{5}(x -(-3))\)

or

\(\displaystyle y - 4 = - \frac{7}{5}(x +3)\)

To find the \(\displaystyle y\)-intercept, substitute 0 for \(\displaystyle x\) and solve for \(\displaystyle y\):

\(\displaystyle y - 4 = - \frac{7}{5}(0+3)\)

\(\displaystyle y - 4 = - \frac{7}{5}(3)\)

\(\displaystyle y - 4 = - \frac{21}{5}\)

\(\displaystyle y - 4+ 4 = - \frac{21}{5} + 4\)

\(\displaystyle y = - \frac{1}{5}\)

The  \(\displaystyle y\)-intercept is \(\displaystyle \left (0, - \frac{1}{5} \right )\).

 

Example Question #498 : Ssat Upper Level Quantitative (Math)

Give the \(\displaystyle x\)-intercept of the line that passes through points \(\displaystyle (-3, 4)\) and \(\displaystyle (2, -3)\).

Possible Answers:

\(\displaystyle \left ( - 8 \frac{1}{5},0 \right )\)

\(\displaystyle \left ( - \frac{1}{5},0 \right )\)

\(\displaystyle \left (- 5\frac{6}{7} , 0 \right )\)

\(\displaystyle \left (- \frac{1}{7} , 0 \right )\)

\(\displaystyle \text{The line has no } x\text{-intercept.}\)

Correct answer:

\(\displaystyle \left (- \frac{1}{7} , 0 \right )\)

Explanation:

First, find the slope of the line, using the slope formula

\(\displaystyle m = \frac{y_{2} -y_{1}}{x_{2}-x_{1}}\)

setting \(\displaystyle x_{1} = -3, y_{1} = 4, x_{2} = 2, y_{2} = -3\):

\(\displaystyle m = \frac{-3 -4}{2-(-3)} = \frac{-7}{5} =- \frac{7}{5}\)

By the point-slope formula, this line has the equation

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

where

\(\displaystyle x_{1} = -3, y_{1} = 4, m = - \frac{7}{5}\); the line becomes

\(\displaystyle y - 4 = - \frac{7}{5}(x -(-3))\)

or

\(\displaystyle y - 4 = - \frac{7}{5}(x +3)\)

To find the \(\displaystyle x\)-intercept, substitute 0 for \(\displaystyle y\) and solve for \(\displaystyle x\):

\(\displaystyle 0 - 4 = - \frac{7}{5}(x +3)\)

\(\displaystyle - 4 = - \frac{7}{5} x - \frac{21}{5}\)

\(\displaystyle - 4+ \frac{21}{5} = - \frac{7}{5} x - \frac{21}{5}+ \frac{21}{5}\)

\(\displaystyle \frac{1}{5} = - \frac{7}{5} x\)

\(\displaystyle - \frac{5}{7} \cdot \frac{1}{5} = - \frac{5}{7}\left ( - \frac{7}{5} x \right )\)

\(\displaystyle - \frac{1}{7} =x\)

The  \(\displaystyle x\)-intercept is \(\displaystyle \left (- \frac{1}{7} , 0 \right )\).

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