Let’s solve the given system of equations step-by-step.
The given system of equations is:
[tex]\[ x + \frac{6}{y} - 7 = 0 \][/tex]
[tex]\[ 2xy + 4 = 10y \][/tex]
Step 1: Solve the first equation for [tex]\( x \)[/tex]
Rewrite the first equation as:
[tex]\[ x + \frac{6}{y} = 7 \][/tex]
Subtract [tex]\(\frac{6}{y}\)[/tex] from both sides:
[tex]\[ x = 7 - \frac{6}{y} \][/tex]
Step 2: Substitute [tex]\( x \)[/tex] from the first equation into the second equation
The second equation is:
[tex]\[ 2xy + 4 = 10y \][/tex]
Substitute [tex]\( x = 7 - \frac{6}{y} \)[/tex] into the second equation:
[tex]\[ 2\left(7 - \frac{6}{y}\right)y + 4 = 10y \][/tex]
Step 3: Simplify and solve for [tex]\( y \)[/tex]
Distribute [tex]\( y \)[/tex]:
[tex]\[ 2(7y - 6) + 4 = 10y \][/tex]
[tex]\[ 14y - 12 + 4 = 10y \][/tex]
Combine like terms:
[tex]\[ 14y - 8 = 10y \][/tex]
Subtract [tex]\( 10y \)[/tex] from both sides:
[tex]\[ 4y - 8 = 0 \][/tex]
Add 8 to both sides:
[tex]\[ 4y = 8 \][/tex]
Divide both sides by 4:
[tex]\[ y = 2 \][/tex]
Step 4: Substitute [tex]\( y \)[/tex] back into the expression for [tex]\( x \)[/tex]
[tex]\[ x = 7 - \frac{6}{y} \][/tex]
Substitute [tex]\( y = 2 \)[/tex]:
[tex]\[ x = 7 - \frac{6}{2} \][/tex]
[tex]\[ x = 7 - 3 \][/tex]
[tex]\[ x = 4 \][/tex]
Therefore, the solution to the system of equations is:
[tex]\[ (x, y) = (4, 2) \][/tex]
