
How do you solve the system \[x=y+4\] and \[2x-5y=2\] by substitution?
Answer
552k+ views
Hint: These types of problems are pretty straight forward and are very easy to solve. For such problems, we need to remember about the theory of linear as well as simultaneous equations. For such problems, we first need to find the value of one parameter in terms of the other and then substitute this parameter in the other given equation to find the value of the other parameter. Once we have found out that value, we can put the value of the found parameter in any of the equations to find out the unknown parameter. Solving simultaneous equation by this method is known as the method of substitution. In our given problem \[x\] and \[y\] are the unknown parameters and we need to solve the equations to find the value.
Complete step by step solution:
Now we start off with the solution to the problem by writing the two equations as,
\[x=y+4\] ---- Equation \[1\]
\[2x-5y=2\] ----- Equation \[2\]
We see that in the first equation that the equation is expressed in terms of an unknown parameter and we need not to rearrange it further. We just put the value of \[x\] from the first equation and put it in the second equation. We hence get,
\[2\left( y+4 \right)-5y=2\]
Now, evaluating and solving we get,
\[\begin{align}
& 2y+8-5y=2 \\
& \Rightarrow -3y=2-8 \\
& \Rightarrow -3y=-6 \\
& \Rightarrow 3y=6 \\
& \Rightarrow y=2 \\
\end{align}\]
Now, putting this value of \[y\] in equation \[1\] we find the value of \[x\] as,
\[\begin{align}
& x=y+4 \\
& \Rightarrow x=2+4 \\
& \Rightarrow x=6 \\
\end{align}\]
Thus the solution to our problem is \[x=6\] and \[y=2\].
Note: For problems like these we need to have an in depth knowledge of linear equations and simultaneous equations. The given problem can also be solved using the method of graphs or graph theory. We first need to arrange the equations in the form of a straight line general form equation and then plot them on the graph paper. The point of intersection of the two lines gives the solution to the problem.
Complete step by step solution:
Now we start off with the solution to the problem by writing the two equations as,
\[x=y+4\] ---- Equation \[1\]
\[2x-5y=2\] ----- Equation \[2\]
We see that in the first equation that the equation is expressed in terms of an unknown parameter and we need not to rearrange it further. We just put the value of \[x\] from the first equation and put it in the second equation. We hence get,
\[2\left( y+4 \right)-5y=2\]
Now, evaluating and solving we get,
\[\begin{align}
& 2y+8-5y=2 \\
& \Rightarrow -3y=2-8 \\
& \Rightarrow -3y=-6 \\
& \Rightarrow 3y=6 \\
& \Rightarrow y=2 \\
\end{align}\]
Now, putting this value of \[y\] in equation \[1\] we find the value of \[x\] as,
\[\begin{align}
& x=y+4 \\
& \Rightarrow x=2+4 \\
& \Rightarrow x=6 \\
\end{align}\]
Thus the solution to our problem is \[x=6\] and \[y=2\].
Note: For problems like these we need to have an in depth knowledge of linear equations and simultaneous equations. The given problem can also be solved using the method of graphs or graph theory. We first need to arrange the equations in the form of a straight line general form equation and then plot them on the graph paper. The point of intersection of the two lines gives the solution to the problem.
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