To solve this problem, we need to set up an equation where the costs from both companies are equal to each other and solve for [tex]\( c \)[/tex], the cost of ink.
Given:
- The cost at Sign Guys is [tex]\( 4.7c + 12 \)[/tex].
- The cost at Signs R Us is [tex]\( 3.4(c + 7) \)[/tex].
Setting these two expressions equal to each other, we have:
[tex]\[ 4.7c + 12 = 3.4(c + 7) \][/tex]
Next, we need to distribute the 3.4 on the right side of the equation:
[tex]\[ 4.7c + 12 = 3.4c + 23.8 \][/tex]
Now, we move all terms involving [tex]\( c \)[/tex] to one side of the equation and constants to the other side. To do this, we subtract [tex]\( 3.4c \)[/tex] from both sides:
[tex]\[ 4.7c - 3.4c + 12 = 23.8 \][/tex]
Simplifying the left side of the equation, we get:
[tex]\[ 1.3c + 12 = 23.8 \][/tex]
Next, we need to isolate [tex]\( c \)[/tex] by moving the constant term on the left side to the right side. We subtract 12 from both sides:
[tex]\[ 1.3c = 23.8 - 12 \][/tex]
Simplifying this, we find:
[tex]\[ 1.3c = 11.8 \][/tex]
To solve for [tex]\( c \)[/tex], we divide both sides by 1.3:
[tex]\[ c = \frac{11.8}{1.3} \][/tex]
[tex]\[ c \approx 9.08 \][/tex]
Now, we use this value to calculate the total cost of printing a sign at either company. Let's use the formula from Sign Guys:
[tex]\[ \text{Total cost} = 4.7c + 12 \][/tex]
[tex]\[ \text{Total cost} = 4.7 \times 9.08 + 12 \][/tex]
[tex]\[ \text{Total cost} \approx 42.676 + 12 \][/tex]
[tex]\[ \text{Total cost} \approx 54.676 \][/tex]
Rounded to the nearest dollar, the total cost is:
[tex]\[ \text{Total cost} \approx \$55 \][/tex]
Therefore, the cost of printing a sign is [tex]\( \$55 \)[/tex].
The answer is [tex]\( \$55 \)[/tex].
