Finding coordinates questions with answers

Questions

1. Question 1Where y = mx + c meets an axis

   Find the coordinates of the point where the line $y = -x + 3$ crosses the $x$-axis.

   Show the worked solution

   1. Step 1: Which coordinate is 0 on the x-axis?Put $y = 0$ into $y = -x + 3$.
   2. Step 2: Solve, then write the point$0 = -x + 3$, so $x = 3$The point is $(3, 0)$.
2. Question 2Where ax + by = c meets an axis

   Find the coordinates of the point where the line $4x + 5y = 10$ crosses the $x$-axis.

   Show the worked solution

   1. Step 1: Which coordinate is 0 on the x-axis?Put $y = 0$ into $4x + 5y = 10$.
   2. Step 2: Solve, then write the point$4x = 10$, so $x = \dfrac{5}{2}$The point is $(\dfrac{5}{2}, 0)$.
3. Question 3Where two lines meet

   Find the coordinates of the point where the lines $y = 2x - 4$ and $y = -x - 1$ meet.

   Show the worked solution

   1. Step 1: Where they meet, the y-values are equal$2x - 4 = -x - 1$$x = 1$
   2. Step 2: Substitute back to find y$y = -2$The lines meet at $(1, -2)$.
4. Question 4A point on a line

   The point $(k, 6)$ lies on the line $y = -4x - 1$. Find $k$.

   Show the worked solution

   1. Step 1: Substitute the point into the equation$6 = -4k - 1$
   2. Step 2: Solve for k$k = -\dfrac{7}{4}$
5. Question 5A point from a given area

   The points $A(1, 0)$ and $B(6, 0)$ lie on the $x$-axis. The point $C$ lies on the line $y = -x - 4$, above the $x$-axis. The area of triangle $ABC$ is $5$ units². Find the coordinates of $C$.

   Show the worked solution

   1. Step 1: The base AB lies along the x-axis: its length$AB = 6 - 1$$AB = 5$
   2. Step 2: Area = ½ × base × height: the height is C's distance from the x-axis$\dfrac{1}{2} \times 5 \times h = 5$$h = \dfrac{2 \times 5}{5}$$h = 2$
   3. Step 3: C is above the x-axis: its y-coordinate$y = 2$
   4. Step 4: C lies on the line: substitute its y-coordinate$2 = -x - 4$$x = -6$
   5. Step 5: Write the coordinates of C$C = (-6, 2)$
6. Question 6Area of a triangle from coordinates

   The vertices of triangle $ABC$ are $A(1, -1)$, $B(1, 2)$ and $C(-2, -3)$. Find the area of triangle $ABC$.

   Show the worked solution

   1. Step 1: AB is parallel to the y-axis: its length is the difference of the y-coordinates$AB = 2 - (-1)$$AB = 3$
   2. Step 2: The height is the distance from C to AB, across: the difference of the x-coordinates$\text{Height} = 1 - (-2)$$\text{Height} = 3$
   3. Step 3: Area = ½ × base × height$\text{Area} = \dfrac{1}{2} \times 3 \times 3$$\phantom{\text{Area}} = 4.5 \text{ units}^2$