solve the equation. check your solution.
11. $x + 7 = 12$
12. $y + 9 = 0$
13. $-z=z + 6$
14. $a - 5 = 8$
15. $b - 14=-3$
16. $37 = c - 29$
17. $21 + m = 4$
18. $n - 72 = 72$
19. $p - 24=-53$
20. $q + 8 = 57$
21. $r - 23=-6$
22. $28 = g + 28$
23. $-13 + t = 10$
24. $216 = u - 129$
25. $177 = 403+w$
26. rebates the advertised price of a dvd player is $185 after a $30 mail - in rebate. using the verbal model below, write and solve an equation to find the price of the dvd player before the rebate is applied.
price before rebate - rebate amount = price after rebate
27. biology when attacked by a giant hornet, japanese honeybees cluster together to form a ball around the hornet and then generate heat by buzzing. the honeybees can endure temperatures of up to $48^{circ}c$, which is $3^{circ}c$ greater than the hornet can tolerate. find the maximum temperature tolerated by a japanese giant hornet.
28. archaeology the great pyramid in egypt was built around 2560 b.c. over the years, it has lost 30 feet of height off its top and is now 451 feet tall. find the original height of the great pyramid.

Question

Accepted Answer

1. **For the equation $x + 7=12$**:
   - **Explanation**:
     - **Step1: Isolate the variable $x$**
       - Subtract 7 from both sides of the equation $x + 7=12$. According to the subtraction - property of equality, if $a=b$, then $a - c=b - c$. So $x+7 - 7=12 - 7$.
     - **Step2: Simplify both sides**
       - On the left - hand side, $x+(7 - 7)=x$, and on the right - hand side, $12−7 = 5$. So $x = 5$.
     - **Check**:
       - Substitute $x = 5$ into the original equation. The left - hand side is $5+7=12$, which is equal to the right - hand side.
2. **For the equation $y + 9=0$**:
   - **Explanation**:
     - **Step1: Isolate the variable $y$**
       - Subtract 9 from both sides of the equation $y + 9=0$. So $y+9 - 9=0 - 9$.
     - **Step2: Simplify both sides**
       - On the left - hand side, $y+(9 - 9)=y$, and on the right - hand side, $0−9=-9$. So $y=-9$.
     - **Check**:
       - Substitute $y=-9$ into the original equation. The left - hand side is $-9 + 9=0$, which is equal to the right - hand side.
3. **For the equation $a−5 = 8$**:
   - **Explanation**:
     - **Step1: Isolate the variable $a$**
       - Add 5 to both sides of the equation $a−5 = 8$. According to the addition - property of equality, if $a=b$, then $a + c=b + c$. So $a-5 + 5=8 + 5$.
     - **Step2: Simplify both sides**
       - On the left - hand side, $a+( - 5+5)=a$, and on the right - hand side, $8 + 5=13$. So $a = 13$.
     - **Check**:
       - Substitute $a = 13$ into the original equation. The left - hand side is $13−5 = 8$, which is equal to the right - hand side.
4. **For the equation $b−14=-3$**:
   - **Explanation**:
     - **Step1: Isolate the variable $b$**
       - Add 14 to both sides of the equation $b−14=-3$. So $b-14 + 14=-3 + 14$.
     - **Step2: Simplify both sides**
       - On the left - hand side, $b+( - 14+14)=b$, and on the right - hand side, $-3 + 14 = 11$. So $b = 11$.
     - **Check**:
       - Substitute $b = 11$ into the original equation. The left - hand side is $11−14=-3$, which is equal to the right - hand side.
5. **For the equation $21 + m=4$**:
   - **Explanation**:
     - **Step1: Isolate the variable $m$**
       - Subtract 21 from both sides of the equation $21 + m=4$. So $21 + m-21=4 - 21$.
     - **Step2: Simplify both sides**
       - On the left - hand side, $m+(21 - 21)=m$, and on the right - hand side, $4−21=-17$. So $m=-17$.
     - **Check**:
       - Substitute $m=-17$ into the original equation. The left - hand side is $21+( - 17)=4$, which is equal to the right - hand side.
6. **For the equation $n−72 = 72$**:
   - **Explanation**:
     - **Step1: Isolate the variable $n$**
       - Add 72 to both sides of the equation $n−72 = 72$. So $n-72 + 72=72 + 72$.
     - **Step2: Simplify both sides**
       - On the left - hand side, $n+( - 72+72)=n$, and on the right - hand side, $72 + 72 = 144$. So $n = 144$.
     - **Check**:
       - Substitute $n = 144$ into the original equation. The left - hand side is $144−72 = 72$, which is equal to the right - hand side.
7. **For the equation $p−24=-53$**:
   - **Explanation**:
     - **Step1: Isolate the variable $p$**
       - Add 24 to both sides of the equation $p−24=-53$. So $p-24 + 24=-53 + 24$.
     - **Step2: Simplify both sides**
       - On the left - hand side, $p+( - 24+24)=p$, and on the right - hand side, $-53 + 24=-29$. So $p=-29$.
     - **Check**:
       - Substitute $p=-29$ into the original equation. The left - hand side is $-29−24=-53$, which is equal to the right - hand side.
8. **For the equation $q + 8=57$**:
   - **Explanation**:
     - **Step1: Isolate the variable $q$**
       - Subtract 8 from both sides of the equation $q + 8=57$. So $q+8 - 8=57 - 8$.
     - **Step2: Simplify both sides**
       - On the left - hand side, $q+(8 - 8)=q$, and on the right - hand side, $57−8 = 49$. So $q = 49$.
     - **Check**:
       - Substitute $q = 49$ into the original equation. The left - hand side is $49+8=57$, which is equal to the right - hand side.
9. **For the equation $r−23=-6$**:
   - **Explanation**:
     - **Step1: Isolate the variable $r$**
       - Add 23 to both sides of the equation $r−23=-6$. So $r-23 + 23=-6 + 23$.
     - **Step2: Simplify both sides**
       - On the left - hand side, $r+( - 23+23)=r$, and on the right - hand side, $-6 + 23 = 17$. So $r = 17$.
     - **Check**:
       - Substitute $r = 17$ into the original equation. The left - hand side is $17−23=-6$, which is equal to the right - hand side.
10. **For the equation $28=g + 28$**:
   - **Explanation**:
     - **Step1: Isolate the variable $g$**
       - Subtract 28 from both sides of the equation $28=g + 28$. So $28-28=g + 28-28$.
     - **Step2: Simplify both sides**
       - On the left - hand side, $28−28 = 0$, and on the right - hand side, $g+(28 - 28)=g$. So $g = 0$.
     - **Check**:
       - Substitute $g = 0$ into the original equation. The left - hand side is $28=0 + 28$, which is equal to the right - hand side.
11. **For the equation $-13 + t=10$**:
   - **Explanation**:
     - **Step1: Isolate the variable $t$**
       - Add 13 to both sides of the equation $-13 + t=10$. So $-13 + t+13=10 + 13$.
     - **Step2: Simplify both sides**
       - On the left - hand side, $t+( - 13+13)=t$, and on the right - hand side, $10 + 13 = 23$. So $t = 23$.
     - **Check**:
       - Substitute $t = 23$ into the original equation. The left - hand side is $-13 + 23=10$, which is equal to the right - hand side.
12. **For the equation $216=u−129$**:
   - **Explanation**:
     - **Step1: Isolate the variable $u$**
       - Add 129 to both sides of the equation $216=u−129$. So $216 + 129=u-129 + 129$.
     - **Step2: Simplify both sides**
       - On the right - hand side, $u+( - 129+129)=u$, and on the left - hand side, $216+129 = 345$. So $u = 345$.
     - **Check**:
       - Substitute $u = 345$ into the original equation. The right - hand side is $345−129 = 216$, which is equal to the left - hand side.
13. **For the equation $177=403 + w$**:
   - **Explanation**:
     - **Step1: Isolate the variable $w$**
       - Subtract 403 from both sides of the equation $177=403 + w$. So $177-403=403 + w-403$.
     - **Step2: Simplify both sides**
       - On the right - hand side, $w+(403 - 403)=w$, and on the left - hand side, $177−403=-226$. So $w=-226$.
     - **Check**:
       - Substitute $w=-226$ into the original equation. The right - hand side is $403+( - 226)=177$, which is equal to the left - hand side.
14. **For problem 26 (Rebates)**:
   - Let $x$ be the price of the DVD player before the rebate.
   - The equation based on the verbal model is $x−30 = 185$.
   - **Explanation**:
     - **Step1: Isolate the variable $x$**
       - Add 30 to both sides of the equation $x−30 = 185$. So $x-30 + 30=185 + 30$.
     - **Step2: Simplify both sides**
       - On the left - hand side, $x+( - 30+30)=x$, and on the right - hand side, $185 + 30 = 215$. So $x = 215$.
     - **Check**:
       - Substitute $x = 215$ into the original equation. The left - hand side is $215−30 = 185$, which is equal to the right - hand side.
15. **For problem 27 (Biology)**:
   - Let $x$ be the maximum temperature tolerated by the Japanese giant hornet.
   - The equation is $x+3 = 48$.
   - **Explanation**:
     - **Step1: Isolate the variable $x$**
       - Subtract 3 from both sides of the equation $x + 3=48$. So $x+3 - 3=48 - 3$.
     - **Step2: Simplify both sides**
       - On the left - hand side, $x+(3 - 3)=x$, and on the right - hand side, $48−3 = 45$. So $x = 45$. The maximum temperature tolerated by the Japanese giant hornet is $45^{\circ}C$.
16. **For problem 28 (Archaeology)**:
   - Let $h$ be the original height of the Great Pyramid.
   - The equation is $h−30 = 451$.
   - **Explanation**:
     - **Step1: Isolate the variable $h$**
       - Add 30 to both sides of the equation $h−30 = 451$. So $h-30 + 30=451 + 30$.
     - **Step2: Simplify both sides**
       - On the left - hand side, $h+( - 30+30)=h$, and on the right - hand side, $451 + 30 = 481$. So $h = 481$ feet.

# Answer:
11. $x = 5$
12. $y=-9$
14. $a = 13$
15. $b = 11$
17. $m=-17$
18. $n = 144$
19. $p=-29$
20. $q = 49$
21. $r = 17$
22. $g = 0$
23. $t = 23$
24. $u = 345$
25. $w=-226$
26. The price before the rebate is $\$215$.
27. The maximum temperature tolerated by the Japanese giant hornet is $45^{\circ}C$.
28. The original height of the Great Pyramid is $481$ feet.

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