Math Homework Solutions

Problem 1

Solve for $x$ :

$2x+3=11$

Steps:

1. Subtract 3 from both sides: $2x=8$
2. Divide both sides by 2: $x=4$

Problem 2

Solve for $y$ :

$3y-5=10$

Steps:

1. Add 5 to both sides: $3y=15$
2. Divide both sides by 3: $y=5$

Problem 3

Solve for $z$ :

$4z+2=18$

Steps:

1. Subtract 2 from both sides: $4z=16$
2. Divide both sides by 4: $z=4$

Problem 4

Solve for $x$ :

$\sqrt{x+1}=3$

Steps:

1. Square both sides: $x+1=9$
2. Subtract 1 from both sides: $x=8$

Problem 5

Solve for $w$ :

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

Steps:

1. Add 2 to both sides: $\frac{3}{4}w=3$
2. Multiply both sides by $\frac{4}{3}$ : $w=4$

Problem 6

Solve for $x$ :

$\frac{1}{2}x+3=7$

Steps:

1. Subtract 3 from both sides: $\frac{1}{2}x=4$
2. Multiply both sides by 2: $x=8$

Problem 7

Solve for $y$ :

$3y^2-12=0$

Steps:

1. Add 12 to both sides: $3y^2=12$
2. Divide both sides by 3: $y^2=4$
3. Take the square root of both sides: $y=\pm 2$

Problem 8

Solve for $z$ :

$4z+\frac{1}{3}=2$

Steps:

1. Subtract $\frac{1}{3}$ from both sides: $4z=2-\frac{1}{3}$
2. Convert $2-\frac{1}{3}$ to a common fraction: $4z=\frac{6}{3}-\frac{1}{3}=\frac{5}{3}$
3. Divide both sides by 4: $z=\frac{5}{12}$

Problem 9

Solve for $x$ :

$\sqrt{x+1}=3$

Steps:

1. Square both sides: $x+1=9$
2. Subtract 1 from both sides: $x=8$

Problem 10

Solve for $w$ :

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

Steps:

1. Add 2 to both sides: $\frac{3}{4}w=3$
2. Multiply both sides by $\frac{4}{3}$ : $w=4$

Problem 11

Solve for $x$ :

$2(x+3)=14$

Steps:

1. Divide both sides by 2: $x+3=7$
2. Subtract 3 from both sides: $x=4$

Problem 12

Solve for $y$ :

$3(y-2)+4=10$

Steps:

1. Subtract 4 from both sides: $3(y-2)=6$
2. Divide both sides by 3: $y-2=2$
3. Add 2 to both sides: $y=4$

Problem 13

Solve for $z$ :

$4(z+1)-3=9$

Steps:

1. Add 3 to both sides: $4(z+1)=12$
2. Divide both sides by 4: $z+1=3$
3. Subtract 1 from both sides: $z=2$

Problem 14

Solve for $w$ :

$\frac{1}{2}(4w-6)=5$

Steps:

1. Multiply both sides by 2: $4w-6=10$
2. Add 6 to both sides: $4w=16$
3. Divide both sides by 4: $w=4$

Problem 15

Solve for $x$ :

$3(2x-5)=21$

Steps:

1. Divide both sides by 3: $2x-5=7$
2. Add 5 to both sides: $2x=12$
3. Divide both sides by 2: $x=6$

Problem 16

Solve for $x$ :

$\sin (x)=\frac{1}{2}$

Steps:

1. Determine the angle(s) $x$ such that $\sin (x)=\frac{1}{2}$ .
2. $x={30}^{\circ }$ or $x={150}^{\circ }$ (or in radians, $x=\frac{\pi }{6}$ or $x=\frac{5\pi }{6}$ ).

Problem 17

Solve for $y$ :

$\cos (y)=\frac{\sqrt{3}}{2}$

Steps:

1. Determine the angle(s) $y$ such that $\cos (y)=\frac{\sqrt{3}}{2}$ .
2. $y={30}^{\circ }$ or $y={330}^{\circ }$ (or in radians, $y=\frac{\pi }{6}$ or $y=\frac{11\pi }{6}$ ).

Problem 18

Solve for $z$ :

$\tan (z)=1$

Steps:

1. Determine the angle(s) $z$ such that $\tan (z)=1$ .
2. $z={45}^{\circ }$ or $z={225}^{\circ }$ (or in radians, $z=\frac{\pi }{4}$ or $z=\frac{5\pi }{4}$ ).

Problem 19

Solve for $w$ :

$2\sin (w)+1=0$

Steps:

1. Subtract 1 from both sides: $2\sin (w)=-1$
2. Divide both sides by 2: $\sin (w)=-\frac{1}{2}$
3. Determine the angle(s) $w$ such that $\sin (w)=-\frac{1}{2}$ .
4. $w={210}^{\circ }$ or $w={330}^{\circ }$ (or in radians, $w=\frac{7\pi }{6}$ or $w=\frac{11\pi }{6}$ ).

Problem 20

Solve for $x$ :

$\cos (2x)=0$

Steps:

1. Determine the angle(s) $2x$ such that $\cos (2x)=0$ .
2. $2x={90}^{\circ }$ or $2x={270}^{\circ }$ (or in radians, $2x=\frac{\pi }{2}$ or $2x=\frac{3\pi }{2}$ ).
3. Divide by 2: $x={45}^{\circ }$ or $x={135}^{\circ }$ (or in radians, $x=\frac{\pi }{4}$ or $x=\frac{3\pi }{4}$ ).

Problem 21

Add the matrices:

$$\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)+\left(\begin{array}{cc}5& 6\\ 7& 8\end{array}\right)$$

Steps:

1. Add corresponding elements: $$\left(\begin{array}{cc}1+5& 2+6\\ 3+7& 4+8\end{array}\right)=\left(\begin{array}{cc}6& 8\\ 10& 12\end{array}\right)$$

Problem 22

Subtract the matrices:

$$\left(\begin{array}{cc}9& 8\\ 7& 6\end{array}\right)-\left(\begin{array}{cc}5& 4\\ 3& 2\end{array}\right)$$

Steps:

1. Subtract corresponding elements: $$\left(\begin{array}{cc}9-5& 8-4\\ 7-3& 6-2\end{array}\right)=\left(\begin{array}{cc}4& 4\\ 4& 4\end{array}\right)$$

Problem 23

Multiply the matrices:

$$\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)\cdot \left(\begin{array}{cc}2& 0\\ 1& 2\end{array}\right)$$

Steps:

1. Calculate each element of the resulting matrix: $$\left(\begin{array}{cc}1\cdot 2+2\cdot 1& 1\cdot 0+2\cdot 2\\ 3\cdot 2+4\cdot 1& 3\cdot 0+4\cdot 2\end{array}\right)=\left(\begin{array}{cc}4& 4\\ 10& 8\end{array}\right)$$

Problem 24

Multiply the matrix by a scalar:

$$3\cdot \left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)$$

Steps:

1. Multiply each element by the scalar: $$\left(\begin{array}{cc}3\cdot 1& 3\cdot 2\\ 3\cdot 3& 3\cdot 4\end{array}\right)=\left(\begin{array}{cc}3& 6\\ 9& 12\end{array}\right)$$

Problem 25

Find the determinant of the matrix:

$$\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)$$

Steps:

1. Use the formula $\text{det}(A)=ad-bc$ : $$\text{det}\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)=(1\cdot 4)-(2\cdot 3)=4-6=-2$$