Certainly! Let's solve the equation step-by-step:
Given:
[tex]\[
\frac{0.4z - 3}{1.5z + 9} = \frac{7}{5}
\][/tex]
Step 1: Cross-Multiply to Eliminate the Fractions
To eliminate the fractions, we can cross-multiply:
[tex]\[
5(0.4z - 3) = 7(1.5z + 9)
\][/tex]
Step 2: Distribute the Constants on Both Sides
Distribute 5 on the left side and 7 on the right side:
[tex]\[
5 \times 0.4z - 5 \times 3 = 7 \times 1.5z + 7 \times 9
\][/tex]
[tex]\[
2z - 15 = 10.5z + 63
\][/tex]
Step 3: Move All Terms Involving [tex]\(z\)[/tex] to One Side
Subtract [tex]\(10.5z\)[/tex] from both sides to get all [tex]\(z\)[/tex]-terms on one side:
[tex]\[
2z - 10.5z - 15 = 63
\][/tex]
[tex]\[
-8.5z - 15 = 63
\][/tex]
Step 4: Isolate the [tex]\(z\)[/tex]-Term
Add 15 to both sides to move the constant term to the right:
[tex]\[
-8.5z = 63 + 15
\][/tex]
[tex]\[
-8.5z = 78
\][/tex]
Step 5: Solve for [tex]\(z\)[/tex]
Divide both sides by [tex]\(-8.5\)[/tex]:
[tex]\[
z = \frac{78}{-8.5}
\][/tex]
Step 6: Simplify the Result
Calculating the division gives us:
[tex]\[
z \approx -9.17647058823529
\][/tex]
Thus, the value of [tex]\(z\)[/tex] is:
[tex]\[
z \approx -9.176
\][/tex]
This completes the detailed, step-by-step solution to the given equation.
