Question 7

We will find points for each line and plot them.

Step 1 — Find points for the first line

Let's consider the first equation. The equation is $x - y + 1 = 0$. We rewrite it to find $y$. $y = x + 1$ We choose some $x$ values to find corresponding $y$ values. If $x = 0$, then $y = 0 + 1 = 1$. So, one point is (0, 1). If $x = 1$, then $y = 1 + 1 = 2$. So, another point is (1, 2). If $x = -1$, then $y = -1 + 1 = 0$. This point is (-1, 0).

Step 2 — Find points for the second line

Let's consider the second equation. The equation is $3x + 2y - 12 = 0$. We rewrite it to find $y$. $2y = 12 - 3x$ $y = \frac{12 - 3x}{2}$ We choose some $x$ values to find corresponding $y$ values. If $x = 0$, then $y = \frac{12 - 3(0)}{2} = \frac{12}{2} = 6$. So, one point is (0, 6). If $x = 2$, then $y = \frac{12 - 3(2)}{2} = \frac{12 - 6}{2} = \frac{6}{2} = 3$. So, another point is (2, 3). If $x = 4$, then $y = \frac{12 - 3(4)}{2} = \frac{12 - 12}{2} = \frac{0}{2} = 0$. This point is (4, 0).

Step 3 — Plot the lines and shade the triangle

We plot the points we found for both lines. Then we draw straight lines through these points. The lines are $x - y + 1 = 0$ and $3x + 2y - 12 = 0$. The $x$-axis is the third side of the triangle. We shade the region enclosed by these three lines.

Step 4 — Determine the vertices of the triangle

Let's find the coordinates of the vertices. Vertex 1 is the intersection of $x - y + 1 = 0$ and the $x$-axis ($y=0$). We substitute $y=0$ into the first equation. $x - 0 + 1 = 0$ $x = -1$ The first vertex is (-1, 0).

Vertex 2 is the intersection of $3x + 2y - 12 = 0$ and the $x$-axis ($y=0$). We substitute $y=0$ into the second equation. $3x + 2(0) - 12 = 0$ $3x - 12 = 0$ $3x = 12$ $x = 4$ The second vertex is (4, 0).

Vertex 3 is the intersection of the two given lines. We use the substitution method. From $x - y + 1 = 0$, we get $y = x + 1$. Substitute this into the second equation: $3x + 2y - 12 = 0$. $3x + 2(x + 1) - 12 = 0$ $3x + 2x + 2 - 12 = 0$ $5x - 10 = 0$ $5x = 10$ $x = 2$ Now we find the $y$-coordinate using $y = x + 1$. $y = 2 + 1$ $y = 3$ The third vertex is (2, 3).

Answer

The coordinates of the vertices of the triangle are: (i) (-1, 0) (ii) (4, 0) (iii) (2, 3)