Common Core: High School - Algebra : Reasoning with Equations & Inequalities

Study concepts, example questions & explanations for Common Core: High School - Algebra

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All Common Core: High School - Algebra Resources

8 Diagnostic Tests 97 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Use Graph Points To Approximate Solutions: Ccss.Math.Content.Hsa Rei.D.11

Where do the following equations intersect?

\(\displaystyle y = - 45 x + 19\)

\(\displaystyle y = 43 x - 4\)

Possible Answers:

\(\displaystyle \left( \frac{23}{88} , \frac{637}{88} \right)\)

\(\displaystyle \left( \frac{64}{47} , \frac{1695}{88} \right)\)

\(\displaystyle \left( \frac{64}{47} , \frac{637}{88} \right)\)

\(\displaystyle \left( - \frac{45}{43} , \frac{637}{88} \right)\)

\(\displaystyle \left( \frac{64}{47} , 13 \right)\)

Correct answer:

\(\displaystyle \left( \frac{23}{88} , \frac{637}{88} \right)\)

Explanation:

To figure out where these equations intersect, we need to set them equal to each other and solve for \(\displaystyle \uptext{x}\).

\(\displaystyle - 45 x + 19 = 43 x - 4\)

\(\displaystyle - 45 x = 43 x - 23\)

\(\displaystyle - 88 x = -23\)

\(\displaystyle x = \frac{23}{88}\)

Now we simply plug this \(\displaystyle \uptext{x}\) value into one of the equations to get the \(\displaystyle \uptext{y}\) value.

\(\displaystyle y = -45 \cdot \frac{23}{88} + 19\)

\(\displaystyle y = - \frac{1035}{88} + 19\)

\(\displaystyle y = \frac{637}{88}\)

Example Question #2 : Use Graph Points To Approximate Solutions: Ccss.Math.Content.Hsa Rei.D.11

Where do the following equations intersect?

\(\displaystyle y = 22 x - 47\)

\(\displaystyle y = - 44 x + 4\)

Possible Answers:

\(\displaystyle \left( \frac{17}{22} , -30 \right)\)

\(\displaystyle \left( - \frac{1}{2} , -30 \right)\)

\(\displaystyle \left( \frac{23}{16} , -65 \right)\)

\(\displaystyle \left( \frac{23}{16} , - \frac{1017}{22} \right)\)

\(\displaystyle \left( \frac{23}{16} , -30 \right)\)

Correct answer:

\(\displaystyle \left( \frac{17}{22} , -30 \right)\)

Explanation:

To figure out where these equations intersect, we need to set them equal to each other and solve for \(\displaystyle \uptext{x}.\)

\(\displaystyle 22 x - 47 = - 44 x + 4\)

\(\displaystyle 22 x = - 44 x + 51\)

\(\displaystyle 66 x = 51\)

\(\displaystyle x = \frac{17}{22}\)

Now we simply plug this \(\displaystyle \uptext{x}\) value into one of the equations to get the \(\displaystyle \uptext{y}\) value.

\(\displaystyle y = 22 \cdot \frac{17}{22} -47\)

\(\displaystyle y = 17 -47\)

\(\displaystyle y = -30\)

 

Example Question #1 : Use Graph Points To Approximate Solutions: Ccss.Math.Content.Hsa Rei.D.11

Where do the following equations intersect?

\(\displaystyle y = 49 x + 17\)

\(\displaystyle y = - 40 x - 23\)

Possible Answers:

\(\displaystyle \left( \frac{32}{17} , \frac{1473}{89} \right)\)

\(\displaystyle \left( \frac{32}{17} , - \frac{447}{89} \right)\)

\(\displaystyle \left( \frac{32}{17} , 3 \right)\)

\(\displaystyle \left( - \frac{40}{89} , - \frac{447}{89} \right)\)

\(\displaystyle \left( - \frac{49}{40} , - \frac{447}{89} \right)\)

Correct answer:

\(\displaystyle \left( - \frac{40}{89} , - \frac{447}{89} \right)\)

Explanation:

To figure out where these equations intersect, we need to set them equal to each other and solve for \(\displaystyle \uptext{x}.\)

Subtract \(\displaystyle 17\) from both sides:

\(\displaystyle 49 x + 17 = - 40 x - 23\)

Add \(\displaystyle 40x\) to both sides:

\(\displaystyle 49 x = - 40 x - 40\)

Divide both sides by \(\displaystyle 89\textup:\)

\(\displaystyle 89 x = -40\)

\(\displaystyle x = - \frac{40}{89}\)

Now plug this \(\displaystyle \uptext{x}\) value into one of the equations to solve for the \(\displaystyle \uptext{y}\) value.

\(\displaystyle y = 49 x + 17\)

\(\displaystyle y = 49 \cdot - \frac{40}{89} + 17\)

Multiply \(\displaystyle \frac{49}{1}\) by \(\displaystyle -\frac{40}{89}\textup:\)

\(\displaystyle y = - \frac{1960}{89} + 17\)

Make fractions with common denominators:

\(\displaystyle \frac{17}{1}=\frac{1513}{89}\)

Add fractions:

\(\displaystyle y = - \frac{1960}{89} + \frac{1513}{89}\)

\(\displaystyle y = - \frac{447}{89}\)

 

Example Question #1 : Use Graph Points To Approximate Solutions: Ccss.Math.Content.Hsa Rei.D.11

Where do the following equations intersect?

\(\displaystyle y = 16 x + 22\)

\(\displaystyle y = - 10 x + 30\)

Possible Answers:

\(\displaystyle \left( \frac{4}{13} , \frac{350}{13} \right)\)

\(\displaystyle \left( - \frac{3}{20} , \frac{350}{13} \right)\)

\(\displaystyle \left( - \frac{3}{20} , 58 \right)\)

\(\displaystyle \left( - \frac{8}{5} , \frac{350}{13} \right)\)

\(\displaystyle \left( - \frac{3}{20} , \frac{290}{13} \right)\)

Correct answer:

\(\displaystyle \left( \frac{4}{13} , \frac{350}{13} \right)\)

Explanation:

To figure out where these equations intersect, we need to set them equal to each other and solve for \(\displaystyle \uptext{x}\).

\(\displaystyle 16 x + 22 = - 10 x + 30\)

Now we subtract \(\displaystyle 22\) from each side.

\(\displaystyle 16 x = - 10 x + 8\)

Now we add \(\displaystyle 10 x\) to each side.

\(\displaystyle 26 x = 8\)

Now we divide by \(\displaystyle 26\) on each side.

\(\displaystyle x = \frac{4}{13}\)

Now we simply plug this \(\displaystyle \uptext{x}\) value into one of the equations to get the \(\displaystyle \uptext{y}\) value.

\(\displaystyle y = 16 \cdot \frac{4}{13} + 22\)

\(\displaystyle y = \frac{64}{13} + 22\)

\(\displaystyle y = \frac{350}{13}\)

 

Example Question #2 : Use Graph Points To Approximate Solutions: Ccss.Math.Content.Hsa Rei.D.11

Where do the following equations intersect?

\(\displaystyle y = - 23 x - 30\)

\(\displaystyle y = 4 x - 13\)

Possible Answers:

\(\displaystyle \left( - \frac{23}{4} , - \frac{419}{27} \right)\)

\(\displaystyle \left( - \frac{7}{17} , -62 \right)\)

\(\displaystyle \left( - \frac{7}{17} , - \frac{827}{27} \right)\)

\(\displaystyle \left( - \frac{7}{17} , - \frac{419}{27} \right)\)

\(\displaystyle \left( - \frac{17}{27} , - \frac{419}{27} \right)\)

Correct answer:

\(\displaystyle \left( - \frac{17}{27} , - \frac{419}{27} \right)\)

Explanation:

To figure out where these equations intersect, we need to set them equal to each other and solve for \(\displaystyle \uptext{x}\).

\(\displaystyle - 23 x - 30 = 4 x - 13\)

Now we add \(\displaystyle 30\) to each side.

\(\displaystyle - 23 x = 4 x + 17\)

Now we subtract \(\displaystyle 4 x\) from each side.

\(\displaystyle - 27 x = 17\)

Now we divide by \(\displaystyle -27\) on each side.

\(\displaystyle x = - \frac{17}{27}\)

Now we simply plug this \(\displaystyle \uptext{x}\) value into one of the equations to get the \(\displaystyle \uptext{y}\) value.

\(\displaystyle y = -23 \cdot - \frac{17}{27} -30\)

\(\displaystyle y = \frac{391}{27} -30\)

\(\displaystyle y = - \frac{419}{27}\)

 

Example Question #3 : Use Graph Points To Approximate Solutions: Ccss.Math.Content.Hsa Rei.D.11


Where do the following equations intersect?

\(\displaystyle y = - 4 x + 20\)

\(\displaystyle y = 12 x - 35\)

Possible Answers:

\(\displaystyle \left( \frac{55}{16} , \frac{25}{4} \right)\)

\(\displaystyle \left( \frac{24}{47} , -7 \right)\)

\(\displaystyle \left( - \frac{1}{3} , \frac{25}{4} \right)\)

\(\displaystyle \left( \frac{24}{47} , \frac{375}{16} \right)\)

\(\displaystyle \left( \frac{24}{47} , \frac{25}{4} \right)\)

Correct answer:

\(\displaystyle \left( \frac{55}{16} , \frac{25}{4} \right)\)

Explanation:

To figure out where these equations intersect, we need to set them equal to each other and solve for \(\displaystyle \uptext{x}\).

\(\displaystyle - 4 x + 20 = 12 x - 35\)

Now we subtract \(\displaystyle 20\) from each side.

\(\displaystyle - 4 x = 12 x - 55\)

Now we subtract \(\displaystyle 12 x\) from each side.

\(\displaystyle - 16 x = -55\)

Now we divide by \(\displaystyle -16\) on each side.

\(\displaystyle x = \frac{55}{16}\)

Now we simply plug this \(\displaystyle \uptext{x}\) value into one of the equations to get the \(\displaystyle \uptext{y}\) value.

\(\displaystyle y = -4 \cdot \frac{55}{16} + 20\)

\(\displaystyle y = - \frac{55}{4} + 20\)

\(\displaystyle y = \frac{25}{4}\)

 

Example Question #4 : Use Graph Points To Approximate Solutions: Ccss.Math.Content.Hsa Rei.D.11

Where do the following equations intersect?

\(\displaystyle y = - 29 x - 22\)

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

Possible Answers:

\(\displaystyle \left( \frac{7}{8} , -61 \right)\)

\(\displaystyle \left( \frac{7}{8} , - \frac{239}{28} \right)\)

\(\displaystyle \left( 29 , - \frac{239}{28} \right)\)

\(\displaystyle \left( \frac{7}{8} , - \frac{629}{28} \right)\)

\(\displaystyle \left( - \frac{13}{28} , - \frac{239}{28} \right)\)

Correct answer:

\(\displaystyle \left( - \frac{13}{28} , - \frac{239}{28} \right)\)

Explanation:

To figure out where these equations intersect, we need to set them equal to each other and solve for \(\displaystyle \uptext{x}\).

\(\displaystyle - 29 x - 22 = - x - 9\)

Now we add \(\displaystyle 22\) to each side.

\(\displaystyle - 29 x = - x + 13\)

Now we add \(\displaystyle x\) to each side.

\(\displaystyle - 28 x = 13\)

Now we divide by \(\displaystyle -28\) on each side.

\(\displaystyle x = - \frac{13}{28}\)

Now we simply plug this \(\displaystyle \uptext{x}\) value into one of the equations to get the \(\displaystyle \uptext{y}\) value.

\(\displaystyle y = -29 \cdot - \frac{13}{28} -22\)

\(\displaystyle y = \frac{377}{28} -22\)

\(\displaystyle y = - \frac{239}{28}\)

 

Example Question #5 : Use Graph Points To Approximate Solutions: Ccss.Math.Content.Hsa Rei.D.11

Where do the following equations intersect?

\(\displaystyle y = 29 x + 30\)

\(\displaystyle y = - 41 x + 33\)

Possible Answers:

\(\displaystyle \left( - \frac{1}{74} , \frac{2187}{70} \right)\)

\(\displaystyle \left( - \frac{29}{41} , \frac{2187}{70} \right)\)

\(\displaystyle \left( - \frac{1}{74} , 51 \right)\)

\(\displaystyle \left( \frac{3}{70} , \frac{2187}{70} \right)\)

\(\displaystyle \left( - \frac{1}{74} , \frac{2103}{70} \right)\)

Correct answer:

\(\displaystyle \left( \frac{3}{70} , \frac{2187}{70} \right)\)

Explanation:

To figure out where these equations intersect, we need to set them equal to each other and solve for \(\displaystyle \uptext{x}\).

\(\displaystyle 29 x + 30 = - 41 x + 33\)

Now we subtract \(\displaystyle 30\) from each side.

\(\displaystyle 29 x = - 41 x + 3\)

Now we add \(\displaystyle 41 x\) to each side.

\(\displaystyle 70 x = 3\)

Now we divide by \(\displaystyle 70\) on each side.

\(\displaystyle x = \frac{3}{70}\)

Now we simply plug this \(\displaystyle \uptext{x}\) value into one of the equations to get the \(\displaystyle \uptext{y}\) value.

\(\displaystyle y = 29 \cdot \frac{3}{70} + 30\)

\(\displaystyle y = \frac{87}{70} + 30\)

\(\displaystyle y = \frac{2187}{70}\)

Example Question #6 : Use Graph Points To Approximate Solutions: Ccss.Math.Content.Hsa Rei.D.11

Where do the following equations intersect?

\(\displaystyle y = - 10 x + 41\)

\(\displaystyle y = 7 x + 7\)

Possible Answers:

\(\displaystyle \left( -\infty , 45 \right)\)

\(\displaystyle \left( - \frac{10}{7} , 21 \right)\)

\(\displaystyle \left( -\infty , 43 \right)\)

\(\displaystyle \left( 2 , 21 \right)\)

\(\displaystyle \left( -\infty , 21 \right)\)

Correct answer:

\(\displaystyle \left( 2 , 21 \right)\)

Explanation:

To figure out where these equations intersect, we need to set them equal to each other and solve for \(\displaystyle \uptext{x}\).

\(\displaystyle - 10 x + 41 = 7 x + 7\)

Now we subtract \(\displaystyle 41\) from each side.

\(\displaystyle - 10 x = 7 x - 34\)

Now we subtract \(\displaystyle 7 x\) from each side.

\(\displaystyle - 17 x = -34\)

Now we divide by \(\displaystyle -17\) on each side.

\(\displaystyle x = 2\)

Now we simply plug this \(\displaystyle \uptext{x}\) value into one of the equations to get the \(\displaystyle \uptext{y}\) value.

\(\displaystyle y = -10 \cdot 2 + 41\)

\(\displaystyle y = -20 + 41\)

\(\displaystyle y = 21\)

 

Example Question #7 : Use Graph Points To Approximate Solutions: Ccss.Math.Content.Hsa Rei.D.11

Where do the following equations intersect?

\(\displaystyle y = - 4 x - 12\)

\(\displaystyle y = - 42 x + 30\)

Possible Answers:

\(\displaystyle \left( \frac{2}{21} , - \frac{312}{19} \right)\)

\(\displaystyle \left( \frac{1}{9} , -28 \right)\)

\(\displaystyle \left( \frac{21}{19} , - \frac{312}{19} \right)\)

\(\displaystyle \left( \frac{1}{9} , - \frac{207}{19} \right)\)

\(\displaystyle \left( \frac{1}{9} , - \frac{312}{19} \right)\)

Correct answer:

\(\displaystyle \left( \frac{21}{19} , - \frac{312}{19} \right)\)

Explanation:

To figure out where these equations intersect, we need to set them equal to each other and solve for \(\displaystyle \uptext{x}\).

\(\displaystyle - 4 x - 12 = - 42 x + 30\)

Now we add \(\displaystyle 12\) to each side.

\(\displaystyle - 4 x = - 42 x + 42\)

Now we add \(\displaystyle 42 x\) to each side.

\(\displaystyle 38 x = 42\)

Now we divide by \(\displaystyle 38\) on each side.

\(\displaystyle x = \frac{21}{19}\)

Now we simply plug this \(\displaystyle \uptext{x}\) value into one of the equations to get the \(\displaystyle \uptext{y}\) value.

\(\displaystyle y = -4 \cdot \frac{21}{19} -12\)

\(\displaystyle y = - \frac{84}{19} -12\)

\(\displaystyle y = - \frac{312}{19}\)

All Common Core: High School - Algebra Resources

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