To solve the simultaneous equations:
[tex]\[
\begin{array}{l}
2x + 4y = 1 \\
3x - 5y = 7
\end{array}
\][/tex]
we can use the method of elimination or substitution to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. Here, I will outline the solution step-by-step using the elimination method.
1. Write down the equations:
[tex]\[2x + 4y = 1 \tag{1}\][/tex]
[tex]\[3x - 5y = 7 \tag{2}\][/tex]
2. Multiply the equations to align coefficients for elimination:
To eliminate [tex]\(y\)[/tex], we can multiply Equation (1) by 5 and Equation (2) by 4:
[tex]\[\begin{aligned}
&5 \cdot (2x + 4y) = 5 \cdot 1 \\
&4 \cdot (3x - 5y) = 4 \cdot 7 \\
\end{aligned} \][/tex]
This results in:
[tex]\[\begin{aligned}
&10x + 20y = 5 \tag{3} \\
&12x - 20y = 28 \tag{4}
\end{aligned} \][/tex]
3. Add the new equations to eliminate [tex]\(y\)[/tex]:
[tex]\[\begin{aligned}
(10x + 20y) + (12x - 20y) &= 5 + 28 \\
10x + 12x &= 33 \\
22x &= 33 \\
x &= \frac{33}{22} \\
x &= \frac{3}{2} \\
x &= 1.5
\end{aligned}\][/tex]
4. Substitute [tex]\(x = 1.5\)[/tex] back into one of the original equations to solve for [tex]\(y\)[/tex]:
Using Equation (1):
[tex]\[\begin{aligned}
2(1.5) + 4y &= 1 \\
3 + 4y &= 1 \\
4y &= 1 - 3 \\
4y &= -2 \\
y &= \frac{-2}{4} \\
y &= -\frac{1}{2} \\
y &= -0.5
\end{aligned}\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[ x = 1.5, \][/tex]
[tex]\[ y = -0.5. \][/tex]
