Question

what is heavy forward but not backward?
shade in the area containing each solution.

1. ( x + 3 ( x + 4 ) = 20 )
2. ( 6 ( y - 1 ) + 8 = 32 )
3. ( 5 + 4 ( n + 9 ) = - 3 )
4. ( 5 ( k - 2 ) - 8 k = - 34 )
5. ( 2 = 11 + 3 ( m + 3 ) )
6. ( - 2 ( p - 5 ) + 7 p = - 5 )
7. ( 4 a - 2 ( a + 9 ) = 6 )
8. ( 7 - 4 ( d - 3 ) = 23 )
9. ( 8 x - 11 ( x - 2 ) = - 8 )
10. ( 5 = 6 ( q - 5 ) - 19 )
11. ( 3 ( 3 - y ) + 1 = 31 )
12. ( 5 ( 2 v + 4 ) = 170 )
13. ( 5 ( 4 n - 4 ) = - 60 )
14. ( 2 ( 3 t - 8 ) - 4 t = 10 )
15. ( 9 - 4 ( 2 p - 1 ) = 45 )
16. write an equation and solve for ( x ) if the area of the rectangle is 70 square units.
17. write an equation and solve for ( x ) if the area of the rectangle is 55 square units.

punchline • bridge to algebra
equations, problems, and functions:
solving equations with parentheses

Explanation:

Step1: Solve \(x + 3(x + 4)=20\)

Expand the equation: \(x+3x + 12=20\). Combine like - terms: \(4x+12 = 20\). Subtract 12 from both sides: \(4x=20 - 12=8\). Divide by 4: \(x = 2\).

Step2: Solve \(6(y - 1)+8 = 32\)

Expand: \(6y-6 + 8=32\). Simplify: \(6y+2 = 32\). Subtract 2: \(6y=30\). Divide by 6: \(y = 5\).

Step3: Solve \(5+4(n + 9)=-3\)

Expand: \(5+4n+36=-3\). Combine like - terms: \(4n + 41=-3\). Subtract 41: \(4n=-44\). Divide by 4: \(n=-11\).

Step4: Solve \(5(k - 2)-8k=-34\)

Expand: \(5k-10-8k=-34\). Combine like - terms: \(-3k-10=-34\). Add 10: \(-3k=-24\). Divide by \(-3\): \(k = 8\).

Step5: Solve \(2=11+3(m + 3)\)

Expand: \(2=11+3m+9\). Combine like - terms: \(2=3m + 20\). Subtract 20: \(-18=3m\). Divide by 3: \(m=-6\).

Step6: Solve \(-2(p - 5)+7p=-5\)

Expand: \(-2p + 10+7p=-5\). Combine like - terms: \(5p+10=-5\). Subtract 10: \(5p=-15\). Divide by 5: \(p=-3\).

Step7: Solve \(4a-2(a + 9)=6\)

Expand: \(4a-2a-18 = 6\). Combine like - terms: \(2a-18=6\). Add 18: \(2a=24\). Divide by 2: \(a = 12\).

Step8: Solve \(7-4(d - 3)=23\)

Expand: \(7-4d+12=23\). Combine like - terms: \(19-4d=23\). Subtract 19: \(-4d=4\). Divide by \(-4\): \(d=-1\).

Step9: Solve \(8x-11(x - 2)=-8\)

Expand: \(8x-11x + 22=-8\). Combine like - terms: \(-3x+22=-8\). Subtract 22: \(-3x=-30\). Divide by \(-3\): \(x = 10\).

Step10: Solve \(5=6(q - 5)-19\)

Expand: \(5=6q-30-19\). Combine like - terms: \(5=6q-49\). Add 49: \(54=6q\). Divide by 6: \(q = 9\).

Step11: Solve \(3(3 - y)+1=31\)

Expand: \(9-3y+1=31\). Combine like - terms: \(10-3y=31\). Subtract 10: \(-3y=21\). Divide by \(-3\): \(y=-7\).

Step12: Solve \(5(2v + 4)=170\)

Expand: \(10v+20 = 170\). Subtract 20: \(10v=150\). Divide by 10: \(v = 15\).

Step13: Solve \(5(4n - 4)=-60\)

Expand: \(20n-20=-60\). Add 20: \(20n=-40\). Divide by 20: \(n=-2\).

Step14: Solve \(2(3t - 8)-4t=10\)

Expand: \(6t-16-4t=10\). Combine like - terms: \(2t-16=10\). Add 16: \(2t=26\). Divide by 2: \(t = 13\).

Step15: Solve \(9-4(2p - 1)=45\)

Expand: \(9-8p + 4=45\). Combine like - terms: \(13-8p=45\). Subtract 13: \(-8p=32\). Divide by \(-8\): \(p=-4\).

Step16: For the rectangle with length \(x + 4\) and width 7 and area 70

Use the area formula \(A=\text{length}\times\text{width}\). So, \(7(x + 4)=70\). Expand: \(7x+28 = 70\). Subtract 28: \(7x=42\). Divide by 7: \(x = 6\).

Step17: For the rectangle with length \(3x-1\) and width 5 and area 55

Use the area formula \(A=\text{length}\times\text{width}\). So, \(5(3x - 1)=55\). Expand: \(15x-5 = 55\). Add 5: \(15x=60\). Divide by 15: \(x = 4\).

Answer:

1. \(x = 2\); 2. \(y = 5\); 3. \(n=-11\); 4. \(k = 8\); 5. \(m=-6\); 6. \(p=-3\); 7. \(a = 12\); 8. \(d=-1\); 9. \(x = 10\); 10. \(q = 9\); 11. \(y=-7\); 12. \(v = 15\); 13. \(n=-2\); 14. \(t = 13\); 15. \(p=-4\); 16. \(x = 6\); 17. \(x = 4\)