Question

solve each equation. check your answer. 1. $5d - 8 = 3 + 7d$ 2. $9 + 2.7t = -4.8t - 6$

Explanation:

Problem 1: Solve \( 5d - 8 = 3 + 7d \)

Step 1: Subtract \( 5d \) from both sides

To get all the \( d \)-terms on one side, we subtract \( 5d \) from both sides of the equation.
\( 5d - 8 - 5d = 3 + 7d - 5d \)
Simplifying both sides, we have:
\( -8 = 3 + 2d \)

Step 2: Subtract 3 from both sides

Now, we want to isolate the term with \( d \). So we subtract 3 from both sides.
\( -8 - 3 = 3 + 2d - 3 \)
Simplifying both sides gives:
\( -11 = 2d \)

Step 3: Divide both sides by 2

To solve for \( d \), we divide both sides of the equation by 2.
\( \frac{-11}{2} = \frac{2d}{2} \)
Simplifying, we get:
\( d = -\frac{11}{2} = -5.5 \)

Step 4: Check the answer

Substitute \( d = -5.5 \) back into the original equation:
Left-hand side (LHS): \( 5(-5.5) - 8 = -27.5 - 8 = -35.5 \)
Right-hand side (RHS): \( 3 + 7(-5.5) = 3 - 38.5 = -35.5 \)
Since LHS = RHS, the solution is correct.

Step 1: Add \( 4.8t \) to both sides

To get all the \( t \)-terms on one side, we add \( 4.8t \) to both sides of the equation.
\( 9 + 2.7t + 4.8t = -4.8t - 6 + 4.8t \)
Simplifying both sides, we have:
\( 9 + 7.5t = -6 \)

Step 2: Subtract 9 from both sides

Now, we isolate the term with \( t \) by subtracting 9 from both sides.
\( 9 + 7.5t - 9 = -6 - 9 \)
Simplifying both sides gives:
\( 7.5t = -15 \)

Step 3: Divide both sides by 7.5

To solve for \( t \), we divide both sides of the equation by 7.5.
\( \frac{7.5t}{7.5} = \frac{-15}{7.5} \)
Simplifying, we get:
\( t = -2 \)

Step 4: Check the answer

Substitute \( t = -2 \) back into the original equation:
Left-hand side (LHS): \( 9 + 2.7(-2) = 9 - 5.4 = 3.6 \)
Right-hand side (RHS): \( -4.8(-2) - 6 = 9.6 - 6 = 3.6 \)
Since LHS = RHS, the solution is correct.

Answer:

(Problem 1): \( d = -5.5 \) (or \( d = -\frac{11}{2} \))

Problem 2: Solve \( 9 + 2.7t = -4.8t - 6 \)