Question

Check whether the pair of equations $x+3y=6$ and $2x-3y=12$ is consistent. If so, solve the equations graphically.
A. Yes, $x=6,y=0$
B. No, $x=6,y=0$
C. Ambiguous
D. Data Insufficient

Answer 1

Hint: First, we will write down the definition of linear equations and then we will write them down condition of linear equation in two variables being consistent that is $\dfrac{{{a}_{1}}}{{{a}_{2}}}\ne \dfrac{{{b}_{1}}}{{{b}_{2}}}$ then we will put the values from the given equations in this condition and get the answer, after that to solve the equations graphically we will first draw the lines on the graph by finding out two points and joining them and then where the two lines interact that will be our solution.

Complete step by step answer:
The equations given to us are: $x+3y=6$ and $2x-3y=12$ are linear equations.
Let’s define what linear equations are, linear equations are those equations that are of the first order. These equations are defined for lines in the coordinate system. A system of linear equations is just a set of two or more linear equations.
 A pair of linear equations in two variables, let’s say: ${{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0\text{ and }{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0$ , now, if equation has only one solution then $\dfrac{{{a}_{1}}}{{{a}_{2}}}\ne \dfrac{{{b}_{1}}}{{{b}_{2}}}$, the linear equations’ pair is called as consistent.

Now, we have with us: $x+3y=6$ and $2x-3y=12$
Where: ${{a}_{1}}=1,\text{ }{{b}_{1}}=3,\text{ }{{c}_{1}}=-6,\text{ }{{a}_{2}}=2,\text{ }{{b}_{2}}=-3,\text{ }{{c}_{2}}=-12$
Now, $\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{1}{2}$ and $\dfrac{{{b}_{1}}}{{{b}_{2}}}=\dfrac{3}{-3}=-1$ , we see that $\dfrac{{{a}_{1}}}{{{a}_{2}}}\ne \dfrac{{{b}_{1}}}{{{b}_{2}}}$ , therefore it has a unique solution.
Hence, the given pair of linear equations is consistent.

Now, we will solve these equations graphically, we will first draw the lines on the graph. For drawing these lines we will first find out the points that lie on this equation
Let’s take: $x+3y=6$, here for $x=0$ , $y=2$ , therefore one point will be: $\left( 0,2 \right)$ , now for $y=0$ , $x=6$ so the next point will be: $\left( 6,0 \right)$. Let’s plot these points on the graph and join them to get the graph of the given line:

Similarly, let’s take: $2x-3y=12$, here for $x=0$ , $y=-4$ , therefore one point will be: $\left( 0,-4 \right)$ , now for $y=0$ , $x=6$ so the next point will be: $\left( 6,0 \right)$. Let’s plot these points on the graph and join them to get the graph of the given line.

From, the graph we can see that the lines intersect only at one point that is: $\left( 6,0 \right)$. Hence the solution of the given pair of the equations will be $x=6\text{ and }y=0$.
In conclusion, the given pair of equations is consistent and have the solution as $x=6\text{ and }y=0$
Hence, option A is correct.
Note:
 Remember that the signs in equations are important, even if you make a mistake even in one place then whole solution will change for example: $2x-3y=12$ , if we write: $-2x+3y=12$ then the solution of : $x+3y=6$ and $-2x+3y=12$ will be: $x=-2,y=\dfrac{8}{3}$ , which totally changes the answer.