Let's solve the equation step by step, detailing the mathematical properties used at each stage. We start with the given equation:
[tex]\[
\frac{5}{c} x + 7 = 22
\][/tex]
Step 1: Subtraction Property of Equality
To isolate the term involving [tex]\( x \)[/tex], we need to get rid of the constant term on the left-hand side. We do this by subtracting 7 from both sides of the equation:
[tex]\[
\frac{5}{c} x + 7 - 7 = 22 - 7
\][/tex]
This simplifies to:
[tex]\[
\frac{5}{c} x = 15
\][/tex]
The reason for this step is the subtraction property of equality, which states that if you subtract the same number from both sides of an equation, the equality remains true.
Step 2: Multiplication Property of Equality
Next, to get rid of the fraction, we multiply both sides of the equation by [tex]\( c \)[/tex]:
[tex]\[
\left(\frac{5}{c} x\right) c = 15 c
\][/tex]
This simplifies to:
[tex]\[
5 x = 15 c
\][/tex]
The reason for this step is the multiplication property of equality, which states that if you multiply both sides of an equation by the same nonzero number, the equality remains true.
Step 3: Division Property of Equality
Finally, to solve for [tex]\( x \)[/tex], we divide both sides of the equation by 5:
[tex]\[
\frac{5 x}{5} = \frac{15 c}{5}
\][/tex]
This simplifies to:
[tex]\[
x = 3 c
\][/tex]
The reason for this step is the division property of equality, which states that if you divide both sides of an equation by the same nonzero number, the equality remains true.
To summarize, the steps and the corresponding reasons are:
1. Statement: [tex]\( \frac{5}{c} x + 7 = 22 \)[/tex] \quad Reason: given
2. Statement: [tex]\( \frac{5}{c} x = 15 \)[/tex] \quad Reason: subtraction property of equality
3. Statement: [tex]\( 5 x = 15 c \)[/tex] \quad Reason: multiplication property of equality
4. Statement: [tex]\( x = 3 c \)[/tex] \quad Reason: division property of equality
This provides a detailed, step-by-step solution to the given equation.
