Question

1. -2(x + 1)+8=-14
2. 10 - 31(6t - 9)=1
3. 2(4n + 1)-3n=17
4. 7 - 3(2c - 5)+c=58
5. -6x + 4(x - 6)=1
6. 9(y + 2)-4y+1=49
7. 4 - 6(v + 3)+2v=-2

Explanation:

1. Solve -2(x + 1)+8=-14:

• Step1: Expand the left - hand side
• Using the distributive property \(a(b + c)=ab+ac\), we have \(-2(x + 1)=-2x-2\). So the equation becomes \(-2x-2 + 8=-14\).
• Simplify the left - hand side: \(-2x+6=-14\).
• Step2: Isolate the term with x
• Subtract 6 from both sides of the equation: \(-2x+6 - 6=-14 - 6\).
• This gives \(-2x=-20\).
• Step3: Solve for x
• Divide both sides by -2: \(\frac{-2x}{-2}=\frac{-20}{-2}\).
• So \(x = 10\).

1. Solve 10-31(6t - 9)=1:

• Step1: Expand the left - hand side
• Using the distributive property, \(-31(6t - 9)=-186t+279\). So the equation becomes \(10-186t + 279=1\).
• Combine like terms: \(-186t+289 = 1\).
• Step2: Isolate the term with t
• Subtract 289 from both sides: \(-186t+289 - 289=1 - 289\).
• We get \(-186t=-288\).
• Step3: Solve for t
• Divide both sides by -186: \(t=\frac{-288}{-186}=\frac{48}{31}\).

1. Solve 2(4n + 1)-3n=17:

• Step1: Expand the left - hand side
• Using the distributive property, \(2(4n + 1)=8n+2\). So the equation becomes \(8n+2-3n=17\).
• Combine like terms: \(5n+2 = 17\).
• Step2: Isolate the term with n
• Subtract 2 from both sides: \(5n+2 - 2=17 - 2\).
• We have \(5n = 15\).
• Step3: Solve for n
• Divide both sides by 5: \(n=\frac{15}{5}=3\).

1. Solve 7-3(2c - 5)+c=58:

• Step1: Expand the left - hand side
• Using the distributive property, \(-3(2c - 5)=-6c + 15\). So the equation becomes \(7-6c+15 + c=58\).
• Combine like terms: \(-5c+22 = 58\).
• Step2: Isolate the term with c
• Subtract 22 from both sides: \(-5c+22 - 22=58 - 22\).
• We get \(-5c=36\).
• Step3: Solve for c
• Divide both sides by -5: \(c=-\frac{36}{5}=-7.2\).

1. Solve -6x+4(x - 6)=1:

• Step1: Expand the left - hand side
• Using the distributive property, \(4(x - 6)=4x-24\). So the equation becomes \(-6x+4x-24 = 1\).
• Combine like terms: \(-2x-24 = 1\).
• Step2: Isolate the term with x
• Add 24 to both sides: \(-2x-24 + 24=1 + 24\).
• We have \(-2x=25\).
• Step3: Solve for x
• Divide both sides by -2: \(x=-\frac{25}{2}=-12.5\).

1. Solve 9(y + 2)-4y+1=49:

• Step1: Expand the left - hand side
• Using the distributive property, \(9(y + 2)=9y+18\). So the equation becomes \(9y+18-4y + 1=49\).
• Combine like terms: \(5y+19 = 49\).
• Step2: Isolate the term with y
• Subtract 19 from both sides: \(5y+19 - 19=49 - 19\).
• We get \(5y = 30\).
• Step3: Solve for y
• Divide both sides by 5: \(y=\frac{30}{5}=6\).

1. Solve 4-6(v + 3)+2v=-2:

• Step1: Expand the left - hand side
• Using the distributive property, \(-6(v + 3)=-6v-18\). So the equation becomes \(4-6v-18 + 2v=-2\).
• Combine like terms: \(-4v-14=-2\).
• Step2: Isolate the term with v
• Add 14 to both sides: \(-4v-14 + 14=-2 + 14\).
• We have \(-4v = 12\).
• Step3: Solve for v
• Divide both sides by -4: \(v=\frac{12}{-4}=-3\).

Answer:

1. \(x = 10\)
2. \(t=\frac{48}{31}\)
3. \(n = 3\)
4. \(c=-7.2\)
5. \(x=-12.5\)
6. \(y = 6\)
7. \(v=-3\)