Question

The graphs of $y-2x=1,4x+y=7,\text{and}2y-x=-4$ contain the sides of a triangle. Find the coordinates of the vertices of the triangle

Short answer

The coordinates of the vertices of the triangle are$\left(1,3\right),\left(-2,-3\right),\left(2,-1\right)$.

Step-by-step solution

Step-1 – Apply the concept of slope-intercept form

Equation of line in slope intercept form is expressed below.

$y=mx+c$

Where m is the slope and c is the intercept of y-axis.

Step-2 –Write the equations in slope-intercept form

The intersection points of the equations represent the vertices of the triangle.

Consider the equations.

$y-2x=1,4x+y=7,\text{and}2y-x=-4$

Consider the first equation$y-2x=1$.

Rewrite the equation in form of slope-intercept form.

Add both sides $2x$

$\begin{array}{l}y-2x+2x=1+2x\\ y=1+2x\\ y=2x+1\end{array}$

Now, the equation is in the form $y=mx+c$. Here slope m of the line is $2$ and intercept of y-axis c is $1$.

Now, consider the second equation$4x+y=7$.

Rewrite the equation in form of slope-intercept form.

Subtract both sides by $4x$

$\begin{array}{l}4x+y-4x=7-4x\\ y=7-4x\\ y=-4x+7\end{array}$

Now, the equation is in the form$y=mx+c$. Here slope m of the line is $-4$ and intercept of y-axis c is 7.

Now, consider the third equation$2y-x=-4$.

Rewrite the equation in form of slope-intercept form.

Add both sides$x$

$\begin{array}{l}2y-x+x=-4+x\\ 2y=x-4\end{array}$

Divide both sides by 2.

$y=\frac{1}{2}x-2$

Now, the equation is in the form $y=mx+c$. Here slope m of the line is $\frac{1}{2}$ and intercept of y-axis c is $-2$

Step-3 – Identify the point of intersection of the equations

Plot the equations on the same plane and the point where both the equations intersect is the solution of the system of the equations.

The red line denotes the equation $y-2x=1$, blue line denotes the equation$4x+y=7$and green line denotes the equation $2y-x=-4$.

Therefore, the point of intersections are $\left(1,3\right),\left(-2,-3\right),\left(2,-1\right)$.

Thus, the coordinates of vertices of the triangle are$\left(1,3\right),\left(-2,-3\right),\left(2,-1\right)$.