Question

determine the value of the unknown in each equation.

1. $2(3 - h) - 6 = -5h$
2. $7 + 9d = 7d + 3$
3. $-2(4 + 3y) = -2(4 + y)$
4. $-7 + 4c = 7c + 6$
5. $5(1 + s) = -9s + 6$
6. $3 + v = 2(2v - 1)$
7. $-2 - 4w = 7w - 8$
8. $-6(1 - m) = 9 - 2m$
9. $-2q - 3 = -2(2q + 1)$
10. $6n + 7 = 2n + 5$
11. $2(3x - 2) + 9 = -5x$
12. $3(1 + p) = -5(p + 1)$
13. $3(1 - 3g) = -7 + g$
14. $1 + 2b = 4b + 9$
15. $2z + 6 = 3z + 1$
16. $5a - 2 = -9a + 8$
17. $6t - 5 = -9t - 9$
18. $-1 + 3f = -7 - 6f$
19. $2 + r = 7 + 6r$
20. $-6k + 1 = -2 + 7k$

Explanation:

Step1: Solve equation 1: \(2(3 - h) - 6 = -5h\)

First, expand the left - hand side: \(2\times3-2h - 6=-5h\), which simplifies to \(6 - 2h-6=-5h\). Then, combine like terms: \(-2h=-5h\). Add \(5h\) to both sides: \(-2h + 5h=-5h + 5h\), so \(3h = 0\), and \(h = 0\).

Step2: Solve equation 2: \(7 + 9d = 7d+3\)

Subtract \(7d\) from both sides: \(7 + 9d-7d=7d + 3-7d\), which gives \(7 + 2d=3\). Then subtract 7 from both sides: \(7 + 2d-7=3 - 7\), so \(2d=-4\), and \(d=-2\).

Step3: Solve equation 3: \(-2(4 + 3y)=-2(4 + y)\)

Expand both sides: \(-8-6y=-8 - 2y\). Add \(6y\) to both sides: \(-8-6y + 6y=-8 - 2y+6y\), which gives \(-8=-8 + 4y\). Add 8 to both sides: \(-8 + 8=-8 + 4y+8\), so \(0 = 4y\), and \(y = 0\).

Step4: Solve equation 4: \(-7 + 4c=7c + 6\)

Subtract \(4c\) from both sides: \(-7 + 4c-4c=7c + 6-4c\), which gives \(-7 = 3c+6\). Subtract 6 from both sides: \(-7-6=3c+6 - 6\), so \(-13 = 3c\), and \(c=-\frac{13}{3}\).

Step5: Solve equation 5: \(5(1 + s)=-9s + 6\)

Expand the left - hand side: \(5+5s=-9s + 6\). Add \(9s\) to both sides: \(5+5s + 9s=-9s + 6+9s\), which gives \(5 + 14s=6\). Subtract 5 from both sides: \(5 + 14s-5=6 - 5\), so \(14s=1\), and \(s=\frac{1}{14}\).

Step6: Solve equation 6: \(3 + v=2(2v - 1)\)

Expand the right - hand side: \(3 + v = 4v-2\). Subtract \(v\) from both sides: \(3 + v-v=4v-2 - v\), which gives \(3 = 3v-2\). Add 2 to both sides: \(3 + 2=3v-2 + 2\), so \(5 = 3v\), and \(v=\frac{5}{3}\).

Step7: Solve equation 7: \(-2-4w=7w - 8\)

Add \(4w\) to both sides: \(-2-4w + 4w=7w - 8+4w\), which gives \(-2 = 11w-8\). Add 8 to both sides: \(-2 + 8=11w-8 + 8\), so \(6 = 11w\), and \(w=\frac{6}{11}\).

Step8: Solve equation 8: \(-6(1 - m)=9 - 2m\)

Expand the left - hand side: \(-6 + 6m=9 - 2m\). Add \(2m\) to both sides: \(-6 + 6m+2m=9 - 2m+2m\), which gives \(-6 + 8m=9\). Add 6 to both sides: \(-6 + 8m+6=9 + 6\), so \(8m=15\), and \(m=\frac{15}{8}\).

Step9: Solve equation 9: \(-2q-3=-2(2q + 1)\)

Expand the right - hand side: \(-2q-3=-4q-2\). Add \(4q\) to both sides: \(-2q-3 + 4q=-4q-2 + 4q\), which gives \(2q-3=-2\). Add 3 to both sides: \(2q-3 + 3=-2 + 3\), so \(2q = 1\), and \(q=\frac{1}{2}\).

Step10: Solve equation 10: \(6n+7 = 2n+5\)

Subtract \(2n\) from both sides: \(6n+7-2n=2n + 5-2n\), which gives \(4n+7 = 5\). Subtract 7 from both sides: \(4n+7-7=5 - 7\), so \(4n=-2\), and \(n=-\frac{1}{2}\).

Step11: Solve equation 11: \(2(3x - 2)+9=-5x\)

Expand the left - hand side: \(6x-4 + 9=-5x\), which simplifies to \(6x + 5=-5x\). Subtract \(6x\) from both sides: \(6x + 5-6x=-5x-6x\), so \(5=-11x\), and \(x=-\frac{5}{11}\).

Step12: Solve equation 12: \(3(1 + p)=-5(p + 1)\)

Expand both sides: \(3 + 3p=-5p-5\). Add \(5p\) to both sides: \(3 + 3p+5p=-5p-5+5p\), which gives \(3 + 8p=-5\). Subtract 3 from both sides: \(3 + 8p-3=-5 - 3\), so \(8p=-8\), and \(p=-1\).

Step13: Solve equation 13: \(3(1 - 3g)=-7 + g\)

Expand the left - hand side: \(3-9g=-7 + g\). Add \(9g\) to both sides: \(3-9g + 9g=-7 + g+9g\), which gives \(3=-7 + 10g\). Add 7 to both sides: \(3 + 7=-7 + 10g+7\), so \(10 = 10g\), and \(g = 1\).

Step14: Solve equation 14: \(1+2b=4b + 9\)

Subtract \(2b\) from both sides: \(1+2b-2b=4b + 9-2b\), which gives \(1 = 2b+9\). Subtract 9 from both sides: \(1-9=2b+9-9\), so \(-8 = 2b\), and \(b=-4\).

Step15: Solve equation 15: \(2z + 6=3z + 1\)

Subtract \(2z\) from both sides: \(2z + 6-2z=3z + 1-2z\), which gives \(6=z + 1\). Subtract 1 from both sides: \(6-1=z + 1-1\), so \(z = 5\).

Step16: Solve equation 16: \(5a-2=-9a + 8\)

Add \(9a\) to both sides: \(5a-2+9a=-9a + 8+9a\), which gives \(14a-2 = 8\). Add 2 to both sides: \(14a-2+2=8 + 2\), so \(14a = 10\), and \(a=\frac{5}{7}\).

Step17: Solve equation 17: \(6t-5=-9t-9\)

Add \(9t\) to both sides: \(6t-5+9t=-9t-9+9t\), which gives \(15t-5=-9\). Add 5 to both sides: \(15t-5+5=-9 + 5\), so \(15t=-4\), and \(t=-\frac{4}{15}\).

Step18: Solve equation 18: \(-1+3f=-7-6f\)

Add \(6f\) to both sides: \(-1+3f+6f=-7-6f+6f\), which gives \(-1 + 9f=-7\). Add 1 to both sides: \(-1 + 9f+1=-7 + 1\), so \(9f=-6\), and \(f=-\frac{2}{3}\).

Step19: Solve equation 19: \(2 + r=7 + 6r\)

Subtract \(r\) from both sides: \(2 + r-r=7 + 6r-r\), which gives \(2 = 7+5r\). Subtract 7 from both sides: \(2-7=7+5r-7\), so \(-5 = 5r\), and \(r=-1\).

Step20: Solve equation 20: \(-6k + 1=-2+7k\)

Add \(6k\) to both sides: \(-6k + 1+6k=-2+7k+6k\), which gives \(1=-2 + 13k\). Add 2 to both sides: \(1 + 2=-2 + 13k+2\), so \(3 = 13k\), and \(k=\frac{3}{13}\).

Answer:

1. \(h = 0\)
2. \(d=-2\)
3. \(y = 0\)
4. \(c=-\frac{13}{3}\)
5. \(s=\frac{1}{14}\)
6. \(v=\frac{5}{3}\)
7. \(w=\frac{6}{11}\)
8. \(m=\frac{15}{8}\)
9. \(q=\frac{1}{2}\)
10. \(n=-\frac{1}{2}\)
11. \(x=-\frac{5}{11}\)
12. \(p=-1\)
13. \(g = 1\)
14. \(b=-4\)
15. \(z = 5\)
16. \(a=\frac{5}{7}\)
17. \(t=-\frac{4}{15}\)
18. \(f=-\frac{2}{3}\)
19. \(r=-1\)
20. \(k=\frac{3}{13}\)