Question

solve the equation. check your solution.

1. -5(2w + 1) = 25
2. 4(5 - p) = 8
3. -40 - (2x + 5) = -61
4. 2 = 4(3k - 8) - 11k
5. 42 = 18t + 4(t + 5)
6. -3(2z - 8) + 10z = 16
7. -5g - (8 - g) = 12
8. -5 = 0.25(4 + 20r) - 8r
9. 2m + 0.5(m - 4) = 9
10. -12 = -2h + 0.2(20 - 6h)
11. photograph the perimeter of a rectangular photograph is 22 inches. the length of the photograph is 1 inch more than the width. what are the dimensions of the photograph?

geometry find the value of x for the given triangle, rectangle, or square.

1. perimeter = 40 units
2. perimeter = 22 units
3. perimeter = 104 units
4. perimeter = 32 units

Explanation:

Step1: Solve equation 22

Distribute -5:

$$-5(2w + 1)=25$$

becomes

$$-10w-5 = 25$$

.
Add 5 to both sides:

$$-10w=25 + 5=30$$

.
Divide both sides by -10:

$$w=\frac{30}{-10}=-3$$

.

Step2: Solve equation 23

Distribute 4:

$$4(5 - p)=8$$

becomes

$$20-4p = 8$$

.
Subtract 20 from both sides:

$$-4p=8 - 20=-12$$

.
Divide both sides by -4:

$$p=\frac{-12}{-4}=3$$

.

Step3: Solve equation 24

Simplify the left - hand side:

$$-40-(2x + 5)=-61$$

becomes

$$-40-2x-5=-61$$

, then

$$-2x-45=-61$$

.
Add 45 to both sides:

$$-2x=-61 + 45=-16$$

.
Divide both sides by -2:

$$x=\frac{-16}{-2}=8$$

.

Step4: Solve equation 25

Distribute 4:

$$2 = 4(3k-8)-11k$$

becomes

$$2=12k-32-11k$$

.
Combine like terms:

$$2=k - 32$$

.
Add 32 to both sides:

$$k=2 + 32=34$$

.

Step5: Solve equation 26

Distribute 4:

$$42=18t + 4(t + 5)$$

becomes

$$42=18t+4t + 20$$

.
Combine like terms:

$$42=22t + 20$$

.
Subtract 20 from both sides:

$$22t=42 - 20=22$$

.
Divide both sides by 22:

$$t=\frac{22}{22}=1$$

.

Step6: Solve equation 27

Distribute -3:

$$-3(2z-8)+10z = 16$$

becomes

$$-6z + 24+10z=16$$

.
Combine like terms:

$$4z+24 = 16$$

.
Subtract 24 from both sides:

$$4z=16 - 24=-8$$

.
Divide both sides by 4:

$$z=\frac{-8}{4}=-2$$

.

Step7: Solve equation 28

Simplify the left - hand side:

$$-5g-(8 - g)=12$$

becomes

$$-5g-8 + g=12$$

, then

$$-4g-8 = 12$$

.
Add 8 to both sides:

$$-4g=12 + 8=20$$

.
Divide both sides by -4:

$$g=\frac{20}{-4}=-5$$

.

Step8: Solve equation 29

Distribute 0.25:

$$-5=0.25(4 + 20r)-8r$$

becomes

$$-5 = 1+5r-8r$$

.
Combine like terms:

$$-5=1-3r$$

.
Subtract 1 from both sides:

$$-3r=-5 - 1=-6$$

.
Divide both sides by -3:

$$r=\frac{-6}{-3}=2$$

.

Step9: Solve equation 30

Distribute 0.5:

$$2m+0.5(m - 4)=9$$

becomes

$$2m+0.5m-2 = 9$$

.
Combine like terms:

$$2.5m-2 = 9$$

.
Add 2 to both sides:

$$2.5m=9 + 2=11$$

.
Divide both sides by 2.5:

$$m=\frac{11}{2.5}=\frac{11\times2}{2.5\times2}=\frac{22}{5}=4.4$$

.

Step10: Solve equation 31

Distribute 0.2:

$$-12=-2h+0.2(20 - 6h)$$

becomes

$$-12=-2h + 4-1.2h$$

.
Combine like terms:

$$-12=-3.2h + 4$$

.
Subtract 4 from both sides:

$$-3.2h=-12 - 4=-16$$

.
Divide both sides by -3.2:

$$h=\frac{-16}{-3.2}=5$$

.

Step11: Solve problem 32

Let the width of the photograph be \(w\) inches. Then the length \(l=w + 1\) inches.
The perimeter formula for a rectangle is \(P = 2(l + w)\). Given \(P = 22\) inches.

$$22=2((w + 1)+w)$$

$$22=2(2w + 1)$$

$$22=4w+2$$

$$4w=22 - 2=20$$

$$w = 5$$

inches, and \(l=w + 1=6\) inches.

Step12: Solve problem 33

For a rectangle with perimeter \(P = 40\) units, length \(l=x + 2\) and width \(w = 7\).
The perimeter formula \(P=2(l + w)\) gives \(40=2((x + 2)+7)\).

$$40=2(x + 9)$$

$$40=2x+18$$

$$2x=40 - 18=22$$

$$x = 11$$

units.

Step13: Solve problem 34

For a triangle with perimeter \(P = 22\) units, sides 5, \(x\) and \(x + 1\).

$$P=5+x+(x + 1)$$

$$22=5+x+x + 1$$

$$22=2x+6$$

$$2x=22 - 6=16$$

$$x = 8$$

units.

Step14: Solve problem 35

For a square with perimeter \(P = 104\) units and side length \(s=x + 11\).
The perimeter formula for a square \(P = 4s\) gives \(104=4(x + 11)\).

$$104=4x+44$$

$$4x=104 - 44=60$$

$$x = 15$$

units.

Step15: Solve problem 36

For a rectangle with perimeter \(P = 32\) units, length \(l=2x + 10\) and width \(w=x\).
The perimeter formula \(P=2(l + w)\) gives \(32=2((2x + 10)+x)\).

$$32=2(3x + 10)$$

$$32=6x+20$$

$$6x=32 - 20=12$$

$$x = 2$$

units.

Answer:

1. \(w=-3\)
2. \(p = 3\)
3. \(x = 8\)
4. \(k = 34\)
5. \(t = 1\)
6. \(z=-2\)
7. \(g=-5\)
8. \(r = 2\)
9. \(m = 4.4\)
10. \(h = 5\)
11. Width: 5 inches, Length: 6 inches
12. \(x = 11\) units
13. \(x = 8\) units
14. \(x = 15\) units
15. \(x = 2\) units