Let's solve the given system of equations using the substitution method. The system of equations is:
[tex]\[ \left\{ \begin{array}{l}
2x - 5 = y \\
\frac{x}{4} + \frac{y}{3} = 2 \\
\end{array}\right. \][/tex]
Step 1: Solve the first equation for [tex]\( y \)[/tex].
[tex]\[ 2x - 5 = y \][/tex]
Rewriting this, we get:
[tex]\[ y = 2x - 5 \][/tex]
Step 2: Substitute the expression for [tex]\( y \)[/tex] from the first equation into the second equation.
The second equation is:
[tex]\[ \frac{x}{4} + \frac{y}{3} = 2 \][/tex]
Substitute [tex]\( y = 2x - 5 \)[/tex] into the second equation:
[tex]\[ \frac{x}{4} + \frac{2x - 5}{3} = 2 \][/tex]
Step 3: Solve for [tex]\( x \)[/tex].
To solve the equation, first clear the fractions by finding a common denominator. The common denominator for 4 and 3 is 12. So, we write:
[tex]\[ \frac{3x}{12} + \frac{4(2x - 5)}{12} = 2 \][/tex]
Simplify the numerators:
[tex]\[ \frac{3x + 8x - 20}{12} = 2 \][/tex]
[tex]\[ \frac{11x - 20}{12} = 2 \][/tex]
To get rid of the denominator, multiply both sides by 12:
[tex]\[ 11x - 20 = 24 \][/tex]
Solving for [tex]\( x \)[/tex], we get:
[tex]\[ 11x = 44 \][/tex]
[tex]\[ x = 4 \][/tex]
Step 4: Substitute [tex]\( x = 4 \)[/tex] back into the expression for [tex]\( y \)[/tex].
The expression for [tex]\( y \)[/tex] from Step 1 is:
[tex]\[ y = 2x - 5 \][/tex]
Substitute [tex]\( x = 4 \)[/tex]:
[tex]\[ y = 2(4) - 5 \][/tex]
[tex]\[ y = 8 - 5 \][/tex]
[tex]\[ y = 3 \][/tex]
So, the solution to the system of equations is:
[tex]\[ (x, y) = (4, 3) \][/tex]
