Draw the graph of the equations x - y + 1 = 0 and 3x + 2y - 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis and shade the tri... Draw the graph of the equations x - y + 1 = 0 and 3x + 2y - 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis and shade the triangular region. Read more

Steps to Solve

1. Identify the equations First, identify the two equations that need to be graphed. Let's say you have the following equations:

$$ y = 2x + 1 $$ $$ y = -x + 4 $$

2. Find the intersection point To find the intersection point of the two lines, set the equations equal to each other:

$$ 2x + 1 = -x + 4 $$

Now, solve for $x$:

$$ 2x + x = 4 - 1 $$ $$ 3x = 3 $$ $$ x = 1 $$

Next, substitute $x = 1$ into one of the original equations to find $y$:

$$ y = 2(1) + 1 = 3 $$

So the intersection point is $(1, 3)$.

3. Determine the x-intercepts To find the x-intercepts (where the lines cross the x-axis), set $y = 0$ for both equations.

For the first equation:

$$ 0 = 2x + 1 $$ $$ 2x = -1 $$ $$ x = -\frac{1}{2} $$

So the x-intercept is $\left(-\frac{1}{2}, 0\right)$.

For the second equation:

$$ 0 = -x + 4 $$ $$ x = 4 $$

So the x-intercept is $(4, 0)$.

4. Gather vertices of the triangle Now, the points that form the vertices of the triangle, made by the intersection point and the two x-intercepts are:

5. $(1, 3)$ (the intersection point)

6. $\left(-\frac{1}{2}, 0\right)$ (x-intercept of the first equation)

7. $(4, 0)$ (x-intercept of the second equation)

8. Graph the lines and the triangle Finally, graph the equations and plot the three points:

• Draw the lines $y = 2x + 1$ and $y = -x + 4$
• Mark the points $(1, 3)$, $\left(-\frac{1}{2}, 0\right)$, and $(4, 0)$ on the graph
• Connect these points to form the triangle.