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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

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Example Questions

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Example Question #10 : Solve Quadratic Equations By Inspection, Quadratic Formula, Factoring, Completing The Square, And Taking Square Roots: Ccss.Math.Content.Hsa Rei.B.4b

Solve $- 24 x^{2} + 17 x - 32 = 0$

Possible Answers:

$x = \frac{65}{48} - \frac{11 i}{48} \sqrt{23} , x = - \frac{31}{48} + \frac{11 i}{48} \sqrt{23}$

$x = - \frac{79}{48} - \frac{11 i}{48} \sqrt{23} , x = \frac{977}{48} + \frac{11 i}{48} \sqrt{23}$

$x = \frac{497}{48} - \frac{11 i}{48} \sqrt{23} , x = - \frac{463}{48} + \frac{11 i}{48} \sqrt{23}$

$x = \frac{17}{48} - \frac{11 i}{48} \sqrt{23} , x = \frac{17}{48} + \frac{11 i}{48} \sqrt{23}$

$x = \frac{17}{24} - \frac{11 i}{24} \sqrt{23} , x = - \frac{17}{24} - \frac{11 i}{24} \sqrt{23}$

Correct answer:

$x = \frac{17}{48} - \frac{11 i}{48} \sqrt{23} , x = \frac{17}{48} + \frac{11 i}{48} \sqrt{23}$

Explanation:

We can solve this by using the quadratic formula.

The quadratic formula is

$\frac{ -b \pm\sqrt{ b^{2} - 4 a c }}{ 2 a }$

$\uptext{a}$ , $\uptext{b}$ , and $\uptext{c}$ correspond to coefficients in the quadratic equation, which is

$a x^{2} + b x - 32 = 0$

In this case a = -24 , b = 17 , and c = -32 .

$\frac{ -17 \pm\sqrt{ 17 ^2- 4 \cdot -24 \cdot -32 }}{ 2 \cdot -24 }$

$\frac{ -17 \pm\sqrt{ 289 - 3072 }}{ -48 }$

$\frac{ -17 \pm\sqrt{ -2783 }}{ -48 }$

Since the number inside the square root is negative, we will have an imaginary answer.

We will put an $\uptext{i}$ , outside the square root sign, and then do normal operations.

$\frac{ -17 \pm i \sqrt{ 2783 }}{ -48 }$

Now we split this up into two equations.

$\frac{ -17 + i \sqrt{ 2783 }}{ -48 }$

$\frac{17}{48} + \frac{11 i}{48} \sqrt{23}$

$\frac{ -17 - i \sqrt{ 2783 }}{ -48 }$

$\frac{17}{48} - \frac{11 i}{48} \sqrt{23}$

So our solutions are $x = \frac{17}{48} - \frac{11 i}{48} \sqrt{23}$ and $x = \frac{17}{48} + \frac{11 i}{48} \sqrt{23}$

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Example Question #11 : Solve Quadratic Equations By Inspection, Quadratic Formula, Factoring, Completing The Square, And Taking Square Roots: Ccss.Math.Content.Hsa Rei.B.4b

Solve $27 x^{2} + 3 x + 5 = 0$

Possible Answers:

$x = - \frac{1}{9} - \frac{\sqrt{59} i}{9} , x = \frac{1}{9} - \frac{\sqrt{59} i}{9}$

$x = \frac{179}{18} - \frac{\sqrt{59} i}{18} , x = - \frac{181}{18} + \frac{\sqrt{59} i}{18}$

$x = \frac{17}{18} - \frac{\sqrt{59} i}{18} , x = - \frac{19}{18} + \frac{\sqrt{59} i}{18}$

$x = - \frac{1}{18} - \frac{\sqrt{59} i}{18} , x = - \frac{1}{18} + \frac{\sqrt{59} i}{18}$

$x = - \frac{37}{18} - \frac{\sqrt{59} i}{18} , x = \frac{359}{18} + \frac{\sqrt{59} i}{18}$

Correct answer:

$x = - \frac{1}{18} - \frac{\sqrt{59} i}{18} , x = - \frac{1}{18} + \frac{\sqrt{59} i}{18}$

Explanation:

We can solve this by using the quadratic formula.

The quadratic formula is

$\frac{ -b \pm\sqrt{ b^{2} - 4 a c }}{ 2 a }$

$\uptext{a}$ , $\uptext{b}$ , and $\uptext{c}$ correspond to coefficients in the quadratic equation, which is

$a x^{2} + b x + 5 = 0$

In this case a = 27 , b = 3 , and c = 5 .

$\frac{ -3 \pm\sqrt{ 3 ^2- 4 \cdot 27 \cdot 5 }}{ 2 \cdot 27 }$

$\frac{ -3 \pm\sqrt{ 9 - 540 }}{ 54 }$

$\frac{ -3 \pm\sqrt{ -531 }}{ 54 }$

Since the number inside the square root is negative, we will have an imaginary answer.

We will put an $\uptext{i}$ , outside the square root sign, and then do normal operations.

$\frac{ -3 \pm i \sqrt{ 531 }}{ 54 }$

Now we split this up into two equations.

$\frac{ -3 + i \sqrt{ 531 }}{ 54 }$

$- \frac{1}{18} + \frac{\sqrt{59} i}{18}$

$\frac{ -3 - i \sqrt{ 531 }}{ 54 }$

$- \frac{1}{18} - \frac{\sqrt{59} i}{18}$

So our solutions are $x = - \frac{1}{18} - \frac{\sqrt{59} i}{18}$ and $x = - \frac{1}{18} + \frac{\sqrt{59} i}{18}$

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Example Question #62 : Reasoning With Equations & Inequalities

Solve $12 x^{2} - 13 x + 41 = 0$

Possible Answers:

$x = - \frac{35}{24} - \frac{\sqrt{1799} i}{24} , x = \frac{493}{24} + \frac{\sqrt{1799} i}{24}$

$x = \frac{37}{24} - \frac{\sqrt{1799} i}{24} , x = - \frac{11}{24} + \frac{\sqrt{1799} i}{24}$

$x = \frac{13}{24} - \frac{\sqrt{1799} i}{24} , x = \frac{13}{24} + \frac{\sqrt{1799} i}{24}$

$x = \frac{13}{12} - \frac{\sqrt{1799} i}{12} , x = - \frac{13}{12} - \frac{\sqrt{1799} i}{12}$

$x = \frac{253}{24} - \frac{\sqrt{1799} i}{24} , x = - \frac{227}{24} + \frac{\sqrt{1799} i}{24}$

Correct answer:

$x = \frac{13}{24} - \frac{\sqrt{1799} i}{24} , x = \frac{13}{24} + \frac{\sqrt{1799} i}{24}$

Explanation:

We can solve this by using the quadratic formula.

The quadratic formula is

$\frac{ -b \pm\sqrt{ b^{2} - 4 a c }}{ 2 a }$

$\uptext{a}$ , $\uptext{b}$ , and $\uptext{c}$ correspond to coefficients in the quadratic equation, which is

$a x^{2} + b x + 41 = 0$

In this case a = 12 , b = -13 , and c = 41 .

$\frac{ 13 \pm\sqrt{ -13 ^2- 4 \cdot 12 \cdot 41 }}{ 2 \cdot 12 }$

$\frac{ 13 \pm\sqrt{ 169 - 1968 }}{ 24 }$

$\frac{ 13 \pm\sqrt{ -1799 }}{ 24 }$

Since the number inside the square root is negative, we will have an imaginary answer.

We will put an $\uptext{i}$ , outside the square root sign, and then do normal operations.

$\frac{ 13 \pm i \sqrt{ 1799 }}{ 24 }$

Now we split this up into two equations.

$\frac{ 13 + i \sqrt{ 1799 }}{ 24 }$

$\frac{13}{24} + \frac{\sqrt{1799} i}{24}$

$\frac{ 13 - i \sqrt{ 1799 }}{ 24 }$

$\frac{13}{24} - \frac{\sqrt{1799} i}{24}$

So our solutions are $x = \frac{13}{24} - \frac{\sqrt{1799} i}{24}$ and $x = \frac{13}{24} + \frac{\sqrt{1799} i}{24}$

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Example Question #1 : Solve System Of Equations Using Substitution: Ccss.Math.Content.Hsa Rei.C.5

Solve for $\uptext{x}$ and $\uptext{y}$ . Round your answer to the nearest hundredth.

x + 6 y = 13

14 x - 3 y = -46

Possible Answers:

x = -0.83 , y = 17.03

x = 19.75 , y = 2.55

x = 13.74 , y = -1.12

x = 18.87 , y = -14.64

x = -2.72 , y = 2.62

Correct answer:

x = -2.72 , y = 2.62

Explanation:

The first step to do is solve for a variable. Let's solve for $\uptext{x}$ first.

x + 6 y = 13

Now we subtract 6 y from each side.

x + 6 y - 6 y = 13 - 6 y

Now we divide by

x = - 6 y + 13

Now since we found $\uptext{x}$ , we can plug this into the other equation and find $\uptext{y}$ .

14 x - 3 y = -46

- 84 y + 182 + - 3 y = -46

Now we solve for $\uptext{y}$ .

- 87 y = -228.0

Now we divide by .

y = 2.62068965517241

Now round to the nearest hundredth.

y= 2.62

Now since we have a value for $\uptext{y}$ , we plug this into the first equation and find $\uptext{x}$ .

x + 15.724137931034482 = 13

Now we subtract 15.724137931034482 from each side.

x = -2.724137931034482

Now we divide by

x = -2.72413793103448

Now round to the nearest hundredth.

x= -2.72

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Example Question #2 : Solve System Of Equations Using Substitution: Ccss.Math.Content.Hsa Rei.C.5

Solve for $\uptext{x}$ and $\uptext{y}$ . Round your answer to the nearest hundredth.

- 3 x + 14 y = -95

38 x + 13 y = 28

Possible Answers:

x = -8.58 , y = -15.58

x = -19.72 , y = -9.45

x = -0.9 , y = -17.17

x = 2.85 , y = -6.18

x = -1.24 , y = 9.2

Correct answer:

x = 2.85 , y = -6.18

Explanation:

The first step to do is solve for a variable. Let's solve for $\uptext{x}$ first.

- 3 x + 14 y = -95

Now we subtract 14 y from each side.

- 3 x + 14 y - 14 y = -95 - 14 y

Now we divide by

$x = \frac{14 y}{3} + \frac{95}{3}$

Now since we found $\uptext{x}$ , we can plug this into the other equation and find $\uptext{y}$ .

38 x + 13 y = 28

$\frac{532 y}{3} + \frac{3610}{3} + 13 y = 28$

Now we solve for $\uptext{y}$ .

$\frac{571 y}{3} = -1175.3333333333333$

Now we divide by $\frac{571}{3}$

y = -6.17513134851138

Now round to the nearest hundredth.

y= -6.18

Now since we have a value for $\uptext{y}$ , we plug this into the first equation and find $\uptext{x}$ .

- 3 x + -86.45183887915935 = -95

- 3 x - 86.4518388791594 = -95

Now we add 86.45183887915935 to each side.

- 3 x = -8.548161120840646

Now we divide by

x = 2.84938704028022

Now round to the nearest hundredth.

x= 2.85

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Example Question #3 : Solve System Of Equations Using Substitution: Ccss.Math.Content.Hsa Rei.C.5

Solve for $\uptext{x}$ and $\uptext{y}$ . Round your answer to the nearest hundredth.

16 x + 42 y = -94

14 x - 26 y = -68

Possible Answers:

x = -19.6 , y = 12.87

x = -5.28 , y = -0.23

x = -16.26 , y = 14.15

x = 11.51 , y = -7.77

x = 0.66 , y = -6.54

Correct answer:

x = -5.28 , y = -0.23

Explanation:

The first step to do is solve for a variable. Let's solve for $\uptext{x}$ first.

16 x + 42 y = -94

Now we subtract 42 y from each side.

16 x + 42 y - 42 y = -94 - 42 y

Now we divide by

$x = - \frac{21 y}{8} - \frac{47}{8}$

Now since we found $\uptext{x}$ , we can plug this into the other equation and find $\uptext{y}$ .

14 x - 26 y = -68

$- \frac{147 y}{4} - \frac{329}{4} + - 26 y = -68$

Now we solve for $\uptext{y}$ .

$- \frac{251 y}{4} = 14.25$

Now we divide by $- \frac{251}{4}$ .

y = -0.227091633466135

Now round to the nearest hundredth.

y= -0.23

Now since we have a value for $\uptext{y}$ , we plug this into the first equation and find $\uptext{x}$ .

16 x + -9.53784860557769 = -94

Now we add 9.53784860557769 to each side.

16 x = -84.4621513944223

Now we divide by

x = -5.27888446215139

Now round to the nearest hundredth.

x= -5.28

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Example Question #4 : Solve System Of Equations Using Substitution: Ccss.Math.Content.Hsa Rei.C.5

Solve for $\uptext{x}$ and $\uptext{y}$ . Round your answer to the nearest hundredth.

5 x + 5 y = 80

- 39 x + 8 y = -60

Possible Answers:

x = -1.9 , y = -12.84

x = -11.68 , y = -2.49

x = 17.0 , y = 18.64

x = -1.79 , y = -9.08

x = 4.0 , y = 12.0

Correct answer:

x = 4.0 , y = 12.0

Explanation:

The first step to do is solve for a variable. Let's solve for $\uptext{x}$ first.

5 x + 5 y = 80

Now we subtract 5 y from each side.

5 x + 5 y - 5 y = 80 - 5 y

Now we divide by

x = - y + 16

Now since we found $\uptext{x}$ , we can plug this into the other equation and find $\uptext{y}$ .

- 39 x + 8 y = -60

39 y - 624 + 8 y = -60

Now we solve for $\uptext{y}$ .

47 y = 564.0

Now we divide by .

y = 12.0

Now round to the nearest hundredth.

y= 12.0

Now since we have a value for $\uptext{y}$ , we plug this into the first equation and find $\uptext{x}$ .

5 x + 60.0 = 80

Now we subtract 60.0 from each side.

5 x = 20.0

Now we divide by

x = 4.0

Now round to the nearest hundredth.

x= 4.0

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Example Question #5 : Solve System Of Equations Using Substitution: Ccss.Math.Content.Hsa Rei.C.5

Solve for $\uptext{x}$ and $\uptext{y}$ . Round your answer to the nearest hundredth.

22 x + 11 y = 98

18 x + 46 y = -57

Possible Answers:

x = -7.34 , y = -17.71

x = -10.41 , y = -4.61

x = -9.08 , y = -13.49

x = -13.59 , y = 14.39

x = 6.31 , y = -3.71

Correct answer:

x = 6.31 , y = -3.71

Explanation:

The first step to do is solve for a variable. Let's solve for $\uptext{x}$ first.

22 x + 11 y = 98

Now we subtract 11 y from each side.

22 x + 11 y - 11 y = 98 - 11 y

Now we divide by

$x = - \frac{y}{2} + \frac{49}{11}$

Now since we found $\uptext{x}$ , we can plug this into the other equation and find $\uptext{y}$ .

18 x + 46 y = -57

$- 9 y + \frac{882}{11} + 46 y = -57$

Now we solve for $\uptext{y}$ .

37 y = -137.1818181818182

Now we divide by .

y = -3.70761670761671

Now round to the nearest hundredth.

y= -3.71

Now since we have a value for $\uptext{y}$ , we plug this into the first equation and find $\uptext{x}$ .

22 x + -40.78378378378378 = 98

22 x - 40.7837837837838 = 98

Now we add 40.78378378378378 on each side.

22 x = 138.78378378378378

Now we divide by

x = 6.30835380835381

Now round to the nearest hundredth.

x= 6.31

Report an Error

Example Question #6 : Solve System Of Equations Using Substitution: Ccss.Math.Content.Hsa Rei.C.5

Solve for $\uptext{x}$ and $\uptext{y}$ . Round your answer to the nearest hundredth.

2 x + 45 y = -15

39 x - 22 y = 64

Possible Answers:

x = 7.4 , y = -15.52

x = 19.33 , y = -12.72

x = 0.31 , y = -7.41

x = -6.91 , y = -4.18

x = 1.42 , y = -0.4

Correct answer:

x = 1.42 , y = -0.4

Explanation:

The first step to do is solve for a variable. Let's solve for $\uptext{x}$ first.

2 x + 45 y = -15

Now we subtract 45 y from each side.

2 x + 45 y - 45 y = -15 - 45 y

Now we divide by

$x = - \frac{45 y}{2} - \frac{15}{2}$

Now since we found $\uptext{x}$ , we can plug this into the other equation and find $\uptext{y}$ .

39 x - 22 y = 64

$- \frac{1755 y}{2} - \frac{585}{2} + - 22 y = 64$

Now we solve for $\uptext{y}$ .

$- \frac{1799 y}{2} = 356.5$

Now we divide by $- \frac{1799}{2}$ .

y = -0.39633129516398

Now round to the nearest hundredth.

y= -0.4

Now since we have a value for $\uptext{y}$ , we plug this into the first equation and find $\uptext{x}$ .

2 x + -17.8349082823791 = -15

2 x - 17.8349082823791 = -15

Now we add -17.8349082823791 to each side.

2 x = 2.8349082823791

Now we divide by

x = 1.41745414118955

Now round to the nearest hundredth.

x= 1.42

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Example Question #1 : Solve System Of Equations Using Substitution: Ccss.Math.Content.Hsa Rei.C.5

Solve for $\uptext{x}$ and $\uptext{y}$ . Round your answer to the nearest hundredth.

- 32 x + 22 y = 61

39 x - 34 y = 53

Possible Answers:

x = -12.55 , y = 5.44

x = 12.66 , y = 5.59

x = -9.34 , y = 18.4

x = -6.05 , y = -2.64

x = -14.09 , y = -17.72

Correct answer:

x = -14.09 , y = -17.72

Explanation:

The first step to do is solve for a variable. Let's solve for $\uptext{x}$ first.

- 32 x + 22 y = 61

Now we subtract 22 y from each side.

- 32 x + 22 y - 22 y = 61 - 22 y

Now we divide by

$x = \frac{11 y}{16} - \frac{61}{32}$

Now since we found $\uptext{x}$ , we can plug this into the other equation and find $\uptext{y}$ .

39 x - 34 y = 53

$\frac{429 y}{16} - \frac{2379}{32} + - 34 y = 53$

Now we solve for $\uptext{y}$ .

$- \frac{115 y}{16} = 127.34375$

Now we divide by $- \frac{115}{16}$ .

y = -17.7173913043478

Now round to the nearest hundredth.

y= -17.72

Now since we have a value for $\uptext{y}$ , we plug this into the first equation and find $\uptext{x}$ .

- 32 x + -389.78260869565213 = 61

- 32 x - 389.782608695652 = 61

Now we add 389.78260869565213 to each side.

- 32 x = 450.78260869565213

Now we divide by

x = -14.0869565217391

Now round to the nearest hundredth.

x= -14.09

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

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

All Common Core: High School - Algebra Resources

Example Questions

Example Question #10 : Solve Quadratic Equations By Inspection, Quadratic Formula, Factoring, Completing The Square, And Taking Square Roots: Ccss.Math.Content.Hsa Rei.B.4b

Example Question #11 : Solve Quadratic Equations By Inspection, Quadratic Formula, Factoring, Completing The Square, And Taking Square Roots: Ccss.Math.Content.Hsa Rei.B.4b

Example Question #62 : Reasoning With Equations & Inequalities

Example Question #1 : Solve System Of Equations Using Substitution: Ccss.Math.Content.Hsa Rei.C.5

Example Question #2 : Solve System Of Equations Using Substitution: Ccss.Math.Content.Hsa Rei.C.5

Example Question #3 : Solve System Of Equations Using Substitution: Ccss.Math.Content.Hsa Rei.C.5

Example Question #4 : Solve System Of Equations Using Substitution: Ccss.Math.Content.Hsa Rei.C.5

Example Question #5 : Solve System Of Equations Using Substitution: Ccss.Math.Content.Hsa Rei.C.5

Example Question #6 : Solve System Of Equations Using Substitution: Ccss.Math.Content.Hsa Rei.C.5

Example Question #1 : Solve System Of Equations Using Substitution: Ccss.Math.Content.Hsa Rei.C.5

All Common Core: High School - Algebra Resources

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