Answer
\[\left\{ \left( 2,1 \right) \right\}\]
Work Step by Step
To find the solution of provided system of equation by substitution method use the following steps:
Step1:
The first equation, \[x+3y=5\], is solved for the x in terms of y.
\[\begin{align}
& x+3y=5 \\
& x=5-3y
\end{align}\]
Step2:
Substitute the value of \[x=5-3y\] into the other equation\[4x+5y=13\]. The equation became in one variable y. Solve the equation for y.
\[\begin{align}
& 4x+5y=13 \\
& 4\left( 5-3y \right)+5y=13 \\
& 20-12y+5y=13 \\
& -7y=13-20
\end{align}\]
Simplify the above equation:
\[\begin{align}
& -7y=13-20 \\
& -7y=-7 \\
& y=1
\end{align}\]
Step3:
Now, substitute the value of y obtained in step2, \[x=5-3y\]:
\[\begin{align}
& x=5-3y \\
& x=5-3\cdot 1 \\
& x=5-3 \\
& x=2 \\
\end{align}\]
Step4:
The value of x and y obtained in step2 and step3, is the solution of the provided system of equations.
Hence \[\left\{ \left( 2,1 \right) \right\}\]is the required solution.
Step5:
Now, to verify that the obtained solution is correct, substitute the values of x and y in both equations.
Put \[x=2\text{ and }y=1\]
\[\begin{align}
& x+3y=5 \\
& 2+3\cdot 1=5 \\
& 2+3=5 \\
& 5=5
\end{align}\]
\[\begin{align}
& 4x+5y=13 \\
& 4\cdot 2+5\cdot 1=13 \\
& 8+5=13 \\
& 13=13
\end{align}\]
Since, \[\left( 2,1 \right)\]satisfies both equations, the set \[\left( 2,1 \right)\]is the solution of the provided system of equations.
