Question

Write four solutions to the equation $2x + y = 7$.

Answer 1

Hint: The given equation is a linear equation in two variables. The solution of any linear equation is an ordered pair on the coordinate plane which satisfies the equation or we can say these ordered pairs lie on the graph of the equation. 

To find the solutions of the given equation, we will substitute the value of $x=a$ in the given equation and we will get the corresponding $y=b$ value or Viceversa. We will obtain a solution as $(a,b)$ where $a,b$ are any real numbers.

Complete step-by-step answer:

Given equation is $2x + y = 7$.

We know that the given equations in the problem are of two variables ‘x’ and ‘y’. so, we cannot get exact values of variables as we have only one relation in two variables, it means there will always be an ‘x’ for any ‘y’ and vice-versa will also be true. So, there will be infinite solutions to each equation given in the problem.

So, here we can put any real value of ‘x’ or ‘y’ and hence get another accordance. So, as we need to find four solutions, let us put four different values of ‘x’ and so try to get four values of ‘y’ corresponding to them.

a) Put $x = 0$ in the given equation,

$ \Rightarrow 2\left( 0 \right) + y = 7$

Multiply the terms,

$ \Rightarrow 0 + y = 7$

The addition of 0 doesn’t change the value,

$\therefore y = 7$

So, one solution to the given equation is $x = 0,y = 7$.

b) Put $x = 1$ in the given equation,

$ \Rightarrow 2\left( 1 \right) + y = 7$

Multiply the terms,

$ \Rightarrow 2 + y = 7$

Move the constant on the right side,

$ \Rightarrow y = 7 - 2$

Subtract the value,

$\therefore y = 5$

So, one more solution to the given equation is $x = 1,y = 5$.

c) Put $x = 2$ in the given equation,

$ \Rightarrow 2\left( 2 \right) + y = 7$

Multiply the terms,

$ \Rightarrow 4 + y = 7$

Move the constant on the right side,

$ \Rightarrow y = 7 - 4$

Subtract the value,

$\therefore y = 3$

So, one more solution to the given equation is $x = 2,y = 3$.

d) Put $x = 3$ in the given equation,

$ \Rightarrow 2\left( 3 \right) + y = 7$

Multiply the terms,

$ \Rightarrow 6 + y = 7$

Move the constant on the right side,

$ \Rightarrow y = 7 - 6$

Subtract the value,

$\therefore y = 1$

So, one more solution to the given equation is $x = 3,y = 1$.

Hence, the four solutions of the equation $2x + y = 7$ are $x = 0,y = 7$, $x = 1,y = 5$, $x = 2,y = 3$ and $x = 3,y = 1$.

Note: In the above process, we substituted $x$ value and got the corresponding $y $ value. We can also follow by substituting the $y$ value and get the $x$ value.

The graph of the line $2x+y=7$ with the above solutions is shown below. 

Observe that the solutions are lying on the line $2x+y=7$.