Question

Find the value of k, if x=2, y=1, is a solution of the equation $$2x+3y=k$$. Find two more solutions of the resultant equation.

Answer 1

Hint: We know that if x and y values are given which is a solution of any equation in variable x and y then the values of x and y must satisfy the equation. Similarly, we will substitute x=2, y=1, in the given equation to find the value of k.

Complete step-by-step answer:
We have been given the equation $$2x+3y=k$$ and x=2, y=1, is a solution of the equation.
Since we know that the solution of the equation must satisfy the equation
$$\begin{array}{rl}& \Rightarrow 2\left(2\right)+3\left(1\right)=k\\ & \Rightarrow 4+3=k\\ & \Rightarrow 7=k\end{array}$$
Hence, the value of k is equal to 7.
Now the resultant equation is $$2x+3y=7$$.
In order to find more solutions of the resultant equation we will write the equation as follows:
$$\begin{array}{rl}& \Rightarrow 2x+3y=7\\ & \Rightarrow 2x=7-3y\\ & \Rightarrow x={\displaystyle \frac{7-3y}{2}}\end{array}$$
For y=0, the value of x is as follows:
$$x={\displaystyle \frac{7-3\times 0}{2}}={\displaystyle \frac{7}{2}}$$
Hence, $$x={\displaystyle \frac{7}{2}},y=0$$ is a solution of the resultant equation.
Again, for y=1 the value of x is as follows:
$$x={\displaystyle \frac{7-(3)}{2}}={\displaystyle \frac{7-3}{2}}={\displaystyle \frac{4}{2}}=2$$
Therefore, the value of k is equal to 7 and the two solutions of the resultant equations are $$x={\displaystyle \frac{7}{2}},y=0$$ and $$x=2,y=1$$.

Note: Be careful and take care of the sign while finding the other two solutions for the resultant equation. Also, remember that the solution for the variable will always satisfy the equation when we substitute the values of corresponding variables in the equation. So, we can even cross check once by re-substituting the values of x and y in the equation.