To address this problem, let's proceed step-by-step:

Step 1: Convert the Equations to Slope-Intercept Form

We begin by converting the given linear equations into slope-intercept form $(y = mx + c)$.

1. Equation 1: $2x + 3y = 12$ $3y = -2x + 12$ $y = -\frac{2}{3}x + 4$

2. Equation 2: $x - y = 1$ $y = x - 1$

Step 2: Plot the Equations

Now, let's plot these two lines on the same graph.

• Line 1: $y = -\frac{2}{3}x + 4$

  • Intercepts: $x$-intercept when $y = 0$: $0 = -\frac{2}{3}x + 4 \Rightarrow x = 6$ So, $(6, 0)$.
  • $y$-intercept: $x = 0$, $y = 4$ So, $(0, 4)$.

• Line 2: $y = x - 1$

  • Intercepts: $x$-intercept when $y = 0$: $0 = x - 1 \Rightarrow x = 1$ So, $(1, 0)$.
  • $y$-intercept: $x = 0$, $y = -1$ So, $(0, -1)$.

Step 3: Find the Intersection Point of the Two Lines

To find the coordinates of the intersection, solve the system of equations:

y &= -\frac{2}{3}x + 4, \\ y &= x - 1. \end{aligned}$$ Set the equations equal to each other: $$-\frac{2}{3}x + 4 = x - 1$$ Multiply by 3 to eliminate the fraction: $$-2x + 12 = 3x - 3$$ $$15 = 5x$$ $$x = 3$$ Substitute $$x = 3$$ into one of the original equations: $$y = 3 - 1 = 2$$ So, the intersection point is $$(3, 2)$$. ### Step 4: Determine the Vertices of the Triangle The vertices of the triangle are where the lines intersect the $$x$$-axis and the intersection point itself: - Intersection of Line 1 with $$x$$-axis: $$(6, 0)$$, - Intersection of Line 2 with $$x$$-axis: $$(1, 0)$$, - Intersection of both lines: $$(3, 2)$$. Thus, the vertices are $$(6, 0)$$, $$(1, 0)$$, and $$(3, 2)$$. ### Step 5: Calculate the Area of the Triangle The area of a triangle with vertices $$(x_1, y_1)$$, $$(x_2, y_2)$$, and $$(x_3, y_3)$$ is given by: $$\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$$ Substituting the vertices: $$\text{Area} = \frac{1}{2} \left| 6(0 - 2) + 1(2 - 0) + 3(0 - 0) \right|$$ $$= \frac{1}{2} \left| -12 + 2 \right|$$ $$= \frac{1}{2} \times 10 = 5 \text{ square units}$$ ### Final Result The area of the triangle is 5 square units. The vertices of the triangle are $$(6, 0)$$, $$(1, 0)$$, and $$(3, 2)$$. Would you like a visual representation of this graph?