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