Question

How do you solve $-x+3y=10,x+y=2$ by graphing and classify the system?

Answer 1

We recall the three forms of writing a linear equation the general form $Ax+By+C=0$, or in standard form $Ax+By=D$ can be represented as a line. We recall that we need two points to plot a line. We plot the lines corresponding to the given equation using intercept points (that is putting $x=0,y=0$ seen by one in each of the equation) and find the solution as coordinates of their point of intersection. If the line intersect the system of equations is independent, if they do not the system is parallel and if they coincide the system is dependent.

Complete step by step answer:
We know from the Cartesian coordinate system that every linear equation $Ax+By+C=0$ or in standard form $Ax+By=D$ can be represented as a line. We are given the following equations in the question in standard from

& -x+3y=10.......\left( 1 \right) \\\ & x+y=2...............\left( 2 \right) \\\ \end{aligned}$$ We know that unique solutions to two linear equations are obtained as the coordinates of the point of intersection of the lines corresponding to them. Let us plot line (1). We put $x=0$ and find the $y-$ intercept in equation (1) of line (1) as $$\begin{aligned} & -0+3y=10 \\\ & \Rightarrow 3y=10 \\\ & \Rightarrow y=\dfrac{10}{3} \\\ \end{aligned}$$ We put $y=0$ and find the $x-$ intercept in equation (1) of line (1) as $$\begin{aligned} & -x+3\cdot 0=10 \\\ & \Rightarrow -x=10 \\\ & \Rightarrow x=-10 \\\ \end{aligned}$$ So we got two points for line (1) as $\left( 0,\dfrac{10}{3} \right),\left( -10,0 \right)$. ). We put $x=0$ and find the $y-$ intercept in equation (2) of line (2) as $$\begin{aligned} & 0+y=2 \\\ & \Rightarrow y=2 \\\ \end{aligned}$$ We put $y=0$ and find the $x-$ intercept in equation (2) of line (2) as $$\begin{aligned} & x+y=2 \\\ & \Rightarrow x=2 \\\ & \Rightarrow x=2 \\\ \end{aligned}$$ So we got two points for line (2) as $\left( 0,2 \right),\left( 2,0 \right)$. We plot line (1) and line (2).  We find the point of intersection as $\left( -1,3 \right)$ . So our solution is $x=-1,y=3$ . Since we have unique solution the system is consistent and independent.$$$$ **Note:** We note that if we get a unique solution (when two lines intersect) or infinite solution (two lines coincide) we say the system is consistent. The unique solution is also called independent solution and infinite solution is called dependent solution. If the lines do not intersect which means parallel we get no solution and we call the system inconsistent.