Question

1. solve the following equations if possible.

a. -3x + 7 = 2x - 1 - 3x
b. 1 - 2y + 5 = 4y + 6
c. 3(x + 1) - 8 = 14 - 2(3x - 4)
d. 3x - 2(5x + 3) = 14 - 2x

Explanation:

Step1: Simplify the equation for a.

Combine like - terms on the right - hand side of the equation \(-3x + 7=2x - 1-3x\). The right - hand side simplifies to \(-3x + 7=(2x-3x)-1=-x - 1\). Then add \(3x\) to both sides: \(7 = 2x-1\).

Step2: Solve for \(x\) in equation a.

Add 1 to both sides: \(7 + 1=2x\), so \(8 = 2x\). Divide both sides by 2, we get \(x = 4\).

Step3: Simplify the equation for b.

Combine like - terms on the left - hand side of the equation \(1-2y + 5=4y + 6\). The left - hand side simplifies to \(6-2y=4y + 6\). Add \(2y\) to both sides: \(6=6y + 6\).

Step4: Solve for \(y\) in equation b.

Subtract 6 from both sides: \(0 = 6y\), so \(y = 0\).

Step5: Simplify the equation for c.

Expand both sides of the equation \(3(x + 1)-8=14-2(3x - 4)\). The left - hand side is \(3x+3 - 8=3x - 5\), and the right - hand side is \(14-(6x - 8)=14-6x + 8=22-6x\). So \(3x-5=22-6x\).

Step6: Solve for \(x\) in equation c.

Add \(6x\) to both sides: \(9x-5=22\). Add 5 to both sides: \(9x=27\). Divide both sides by 9, we get \(x = 3\).

Step7: Simplify the equation for d.

Expand the left - hand side of the equation \(3x-2(5x + 3)=14-2x\). The left - hand side is \(3x-(10x + 6)=3x-10x - 6=-7x - 6\). So \(-7x-6=14-2x\).

Step8: Solve for \(x\) in equation d.

Add \(7x\) to both sides: \(-6=14 + 5x\). Subtract 14 from both sides: \(-20=5x\). Divide both sides by 5, we get \(x=-4\).

Answer:

a. \(x = 4\)
b. \(y = 0\)
c. \(x = 3\)
d. \(x=-4\)