Answer :
Sure, let's solve the given system of equations step-by-step. The system of equations is:
[tex]\[ \begin{array}{l} \frac{x}{2} + y = 0.8 \quad \text{(Equation 1)}\\ \frac{7}{x + \frac{y}{2}} = 10 \quad \text{(Equation 2)} \end{array} \][/tex]
Step 1: Solve the first equation for [tex]\( y \)[/tex].
Starting with Equation 1:
[tex]\[ \frac{x}{2} + y = 0.8 \][/tex]
Subtract [tex]\(\frac{x}{2}\)[/tex] from both sides:
[tex]\[ y = 0.8 - \frac{x}{2} \][/tex]
Step 2: Substitute the expression for [tex]\( y \)[/tex] into the second equation.
Using [tex]\( y = 0.8 - \frac{x}{2} \)[/tex] in Equation 2:
[tex]\[ \frac{7}{x + \frac{0.8 - \frac{x}{2}}{2}} = 10 \][/tex]
Simplify the denominator of the fraction:
[tex]\[ \frac{7}{x + \frac{0.8}{2} - \frac{x}{4}} = 10 \][/tex]
[tex]\[ \frac{7}{x + 0.4 - \frac{x}{4}} = 10 \][/tex]
Combine like terms in the denominator:
[tex]\[ \frac{7}{x - \frac{x}{4} + 0.4} = 10 \][/tex]
Simplify the terms involving [tex]\( x \)[/tex]:
[tex]\[ \frac{7}{\frac{4x - x}{4} + 0.4} = 10 \][/tex]
[tex]\[ \frac{7}{\frac{3x}{4} + 0.4} = 10 \][/tex]
Step 3: Solve for [tex]\( x \)[/tex].
To clear the fraction in the denominator, multiply both sides by [tex]\(\frac{3x}{4} + 0.4\)[/tex]:
[tex]\[ 7 = 10 \left( \frac{3x}{4} + 0.4 \right) \][/tex]
Distribute the 10:
[tex]\[ 7 = \frac{30x}{4} + 4 \][/tex]
Simplify the fraction:
[tex]\[ 7 = \frac{15x}{2} + 4 \][/tex]
Subtract 4 from both sides:
[tex]\[ 3 = \frac{15x}{2} \][/tex]
Multiply both sides by [tex]\(\frac{2}{15}\)[/tex]:
[tex]\[ x = \frac{3 \cdot 2}{15} = \frac{6}{15} = \frac{2}{5} = 0.4 \][/tex]
So, [tex]\( x = 0.4 \)[/tex].
Step 4: Substitute [tex]\( x \)[/tex] back into the expression for [tex]\( y \)[/tex].
Using [tex]\( x = 0.4 \)[/tex] in [tex]\( y = 0.8 - \frac{x}{2} \)[/tex]:
[tex]\[ y = 0.8 - \frac{0.4}{2} \][/tex]
[tex]\[ y = 0.8 - 0.2 \][/tex]
[tex]\[ y = 0.6 \][/tex]
Therefore, the solution to the system of equations is:
[tex]\[ x = 0.4 \quad \text{and} \quad y = 0.6 \][/tex]
[tex]\[ \begin{array}{l} \frac{x}{2} + y = 0.8 \quad \text{(Equation 1)}\\ \frac{7}{x + \frac{y}{2}} = 10 \quad \text{(Equation 2)} \end{array} \][/tex]
Step 1: Solve the first equation for [tex]\( y \)[/tex].
Starting with Equation 1:
[tex]\[ \frac{x}{2} + y = 0.8 \][/tex]
Subtract [tex]\(\frac{x}{2}\)[/tex] from both sides:
[tex]\[ y = 0.8 - \frac{x}{2} \][/tex]
Step 2: Substitute the expression for [tex]\( y \)[/tex] into the second equation.
Using [tex]\( y = 0.8 - \frac{x}{2} \)[/tex] in Equation 2:
[tex]\[ \frac{7}{x + \frac{0.8 - \frac{x}{2}}{2}} = 10 \][/tex]
Simplify the denominator of the fraction:
[tex]\[ \frac{7}{x + \frac{0.8}{2} - \frac{x}{4}} = 10 \][/tex]
[tex]\[ \frac{7}{x + 0.4 - \frac{x}{4}} = 10 \][/tex]
Combine like terms in the denominator:
[tex]\[ \frac{7}{x - \frac{x}{4} + 0.4} = 10 \][/tex]
Simplify the terms involving [tex]\( x \)[/tex]:
[tex]\[ \frac{7}{\frac{4x - x}{4} + 0.4} = 10 \][/tex]
[tex]\[ \frac{7}{\frac{3x}{4} + 0.4} = 10 \][/tex]
Step 3: Solve for [tex]\( x \)[/tex].
To clear the fraction in the denominator, multiply both sides by [tex]\(\frac{3x}{4} + 0.4\)[/tex]:
[tex]\[ 7 = 10 \left( \frac{3x}{4} + 0.4 \right) \][/tex]
Distribute the 10:
[tex]\[ 7 = \frac{30x}{4} + 4 \][/tex]
Simplify the fraction:
[tex]\[ 7 = \frac{15x}{2} + 4 \][/tex]
Subtract 4 from both sides:
[tex]\[ 3 = \frac{15x}{2} \][/tex]
Multiply both sides by [tex]\(\frac{2}{15}\)[/tex]:
[tex]\[ x = \frac{3 \cdot 2}{15} = \frac{6}{15} = \frac{2}{5} = 0.4 \][/tex]
So, [tex]\( x = 0.4 \)[/tex].
Step 4: Substitute [tex]\( x \)[/tex] back into the expression for [tex]\( y \)[/tex].
Using [tex]\( x = 0.4 \)[/tex] in [tex]\( y = 0.8 - \frac{x}{2} \)[/tex]:
[tex]\[ y = 0.8 - \frac{0.4}{2} \][/tex]
[tex]\[ y = 0.8 - 0.2 \][/tex]
[tex]\[ y = 0.6 \][/tex]
Therefore, the solution to the system of equations is:
[tex]\[ x = 0.4 \quad \text{and} \quad y = 0.6 \][/tex]
