Question

How do you solve the system of equations:$5x-2y=23;5x+2y=17$ ?

Answer 1

Hint: In this question, we have to find the value of x and y from a system of equations. Therefore, we will use the elimination method to get the result for the solution. In the elimination method, we will eliminate one variable by either adding or subtracting the equations to get an equation in one variable. Thus, we see in both the equations that the coefficient of x is the same, thus we will subtract both the equations. After the necessary calculations, we get the value of y. Thus, we will substitute the value of y in any one of the equations and solve for, which gives the required result for the solution.

Complete step by step solution:
According to the problem, we have to find the value of x and y from the system of equations.
Thus, we will use the elimination method to get the solution.
The equation given to us is $5x-2y=23;5x+2y=17$ .
Let us say that $5x-2y=23$ -------- (1) and $5x+2y=17$ --------- (2)
Now, we see that in both equation (1) and (2), the coefficient of x is the same, thus we will subtract both the equations, that is
$\begin{align}
  & \text{ }5x-2y=23 \\
 & \underset{(-)}{\mathop{{}}}\,\left( 5x+2y=17 \right) \\
\end{align}$
Thus, opening the brackets of the second equation, we get
$\begin{align}
  & \text{ }5x-2y=+\text{ }23 \\
 & \underline{\underset{(-)}{\mathop{+}}\,5x\underset{(-)}{\mathop{+}}\,2y=\underset{(-)}{\mathop{+}}\,17} \\
\end{align}$
As we know, the same terms with opposite signs cancel out each other, thus we get
$\begin{align}
  & \text{ }5x-2y=+\text{ }23 \\
 & \underline{\underset{(-)}{\mathop{+}}\,5x\underset{(-)}{\mathop{+}}\,2y=\underset{(-)}{\mathop{+}}\,17} \\
 & \text{ 0 }-4y=+06 \\
\end{align}$
Now, we will divide 4 on both sides in the above equation, we get
$\begin{align}
 & \text{ 0 }-\dfrac{4}{4}y=+\dfrac{6}{4} \\
\end{align}$
Thus, on further solving, we get
$\begin{align}
   & \text{ }-y=+\dfrac{3}{2} \\
\end{align}$
So, we will multiply (-1) on both sides in the above equation, we get
$-y.(-1)=+\dfrac{3}{2}.(-1)$
Therefore, we get
$y=-\dfrac{3}{2}$
Now, we will substitute the above value of y in equation (1), we get
$5x-2.\left( -\dfrac{3}{2} \right)=23$
On further simplification, we get
$5x+3=23$
Now, we will subtract 3 on both sides in the above equation, we get
$5x+3-3=23-3$
As we know, the same terms with opposite signs cancel out each other, thus we get
$5x=20$
Now, we will divide 5 on both sides in the above equation, we get
$\dfrac{5}{5}x=\dfrac{20}{5}$
On further solving, we get
$x=4$

Therefore, for the system of equations $5x-2y=23;5x+2y=17$ , the value of x and y is 4 and $-\dfrac{3}{2}$ respectively.

Note: While solving this problem, do all the steps properly to avoid confusion and mathematical mistakes. One of the alternative methods to solve this problem is either you can use the substitution method or the cross-multiplication method, to get the required result for the solution.