The co-ordinate diagram below shows the lines q, r, s, and t. q is parallel to s, and r is parallel to t. | Line (q, r, s, or t) | Equation | |----------------------|-------------------------------| | r | y = x - 3 | | t | y = 2x + 3 | | q | y = x + 3 | (b) Use algebra to find the point of intersection of the lines y = x - 3 and y = 2x + 3.

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The co-ordinate diagram below shows the lines q, r, s, and t - Junior Cycle Mathematics - Question 11 - 2017

Question 11

The co-ordinate diagram below shows the lines q, r, s, and t. q is parallel to s, and r is parallel to t. | Line (q, r, s, or t) | Equation | ... show full transcript

Worked Solution & Example Answer:The co-ordinate diagram below shows the lines q, r, s, and t - Junior Cycle Mathematics - Question 11 - 2017

Step 1

Complete the table above to show the equation of each line in the diagram.

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Answer

To complete the table, we need to determine the equations of lines q, r, and t based on their relationships:

Line r: The equation is given as y = x - 3.
Line t: Since it is mentioned that line t is parallel to line r, their slopes are the same. From the equation y = 2x + 3, we have the slope of t as 2. Thus, line t cannot be parallel with line r, re-evaluating:
- The correct slope for line s is 1 (parallel to q), therefore by altering the y-intercept, line s could be represented as y = x + b, where b can be any value, deducing from the drawn parallelism.
Line q: Being parallel to s, its equation can be expressed as y = x + C, correctly finalizing y = x + 3. Thus, the equations are:
- q: y = x + 3
- r: y = x - 3
- t: y = 2x + 3
- s: y = x + 3.

Step 2

Use algebra to find the point of intersection of the lines y = x - 3 and y = 2x + 3.

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Answer

To find the intersection of the two lines:

Start with the equations:
- Line 1: y = x - 3
- Line 2: y = 2x + 3
Equate both equations since at the intersection point, both y-values are equal:
$x - 3 = 2x + 3$
Rearranging yields:
$-3 - 3 = 2x - x$
which simplifies to:
$-6 = x$
Therefore, we have x = -6.
Substitute x = -6 into the first equation to find y:
$y = -6 - 3 = -9$
Thus, the point of intersection is (-6, -9).

Step 3

The line l is a vertical line. It cuts the line y = x - 3 at the point A. It cuts the line y = x + 3 at the point B. Find the distance |AB|.

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Answer

To find the distance between points A and B:

Identify Points A and B:
- Point A from y = x - 3 when x = -6 gives:
  - For line 1:
  $y = -6 - 3 = -9$

ightarrow A(-6,-9) $$

Point B from y = x + 3 when x = -6 gives:
- For line 2:
$y = -6 + 3 = -3$

ightarrow B(-6,-3) $$

Calculate Distance |AB|:
- Use the distance formula:
$|AB| = |y_A - y_B| = |-9 - (-3)| = |-9 + 3| = |-6| = 6$ Thus, the distance |AB| is 6.

The co-ordinate diagram below shows the lines q, r, s, and t - Junior Cycle Mathematics - Question 11 - 2017

Worked Solution & Example Answer:The co-ordinate diagram below shows the lines q, r, s, and t - Junior Cycle Mathematics - Question 11 - 2017

Complete the table above to show the equation of each line in the diagram.

Use algebra to find the point of intersection of the lines y = x - 3 and y = 2x + 3.

The line l is a vertical line. It cuts the line y = x - 3 at the point A. It cuts the line y = x + 3 at the point B. Find the distance |AB|.

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