Combine space c equals 4 invisible times a minus 4 space and space 2 invisible times b equals 5 invisible times a minus 5  to give an equation for c not involving a.

If your answer involves fractions, give the exact value as a simplified fraction, and not a decimal approximation. 

For example, if your answer is 4/3 then write it as 4/3 and not 1.333. (negative 2 b plus 4 over 3)



We are given two equations:

\[
c = 4(a - 4)
\]
\[
2b = 5a - 5
\]

Our goal is to combine these two equations to eliminate \( a \) and express \( c \) in terms of \( b \) only.

### Step 1: Simplify both equations.

1. Expand the first equation for \( c \):
   \[
   c = 4a - 16
   \]

2. Simplify the second equation for \( a \):
   \[
   2b = 5a - 5
   \]
   Solve for \( a \) by isolating it:
   \[
   5a = 2b + 5
   \]
   \[
   a = \frac{2b + 5}{5}
   \]

### Step 2: Substitute the expression for \( a \) into the equation for \( c \).

We now substitute \( a = \frac{2b + 5}{5} \) into the equation for \( c \):

\[
c = 4\left(\frac{2b + 5}{5}\right) - 16
\]

### Step 3: Simplify the expression for \( c \).

Distribute the 4 over the fraction:
\[
c = \frac{4(2b + 5)}{5} - 16
\]
\[
c = \frac{8b + 20}{5} - 16
\]

Now, combine the terms:
\[
c = \frac{8b + 20}{5} - \frac{80}{5}
\]
\[
c = \frac{8b + 20 - 80}{5}
\]
\[
c = \frac{8b - 60}{5}
\]
\[
c = \frac{8(b - 7.5)}{5}
\]


Therefore, the equation for \( c \), not involving \( c\text

Combine space c equals 4 invisible times a minus 4 space and space 2 invisible times b equals 5 invisible times a minus 5 to give an equation for c not involving a. If your answer involves fractions, give the exact value as a simplified fraction, and not a decimal approximation.

This step-by-step guide shows how to combine two linear equations, c = 4(a - 4) and 2b = 5a - 5, to express c in terms of b, using algebraic substitution and simplification techniques. Suitable for students in Grades 9-11.

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