Question

Draw the graphs of the pair of linear equations \(x+3y-6=0\) and \(2x-3y-12=0\) and solve them.

Hint:

To easily find points for graphing, set x=0 to find the y-intercept and then set y=0 to find the x-intercept. These two points are often the simplest to calculate and plot.

Answer 1

Step 1: Understanding the Concept:
To solve a system of linear equations graphically, we need to draw the graph for each equation. The point where the two lines intersect is the solution to the system. To draw a line, we need at least two points that satisfy the equation.

Step 2: Finding Points for Each Line:
For the first equation: \(x + 3y - 6 = 0\)
We can rewrite it as \(x = 6 - 3y\). Let's find two points: \[\begin{array}{rl} \bullet & \text{If \(y = 0\), then \(x = 6 - 3(0) = 6\). So, one point is A(6, 0).} \\ \bullet & \text{If \(y = 2\), then \(x = 6 - 3(2) = 0\). So, another point is B(0, 2).} \\ \end{array}\] For the second equation: \(2x - 3y - 12 = 0\)
We can rewrite it as \(2x = 12 + 3y\). Let's find two points: \[\begin{array}{rl} \bullet & \text{If \(y = 0\), then \(2x = 12 + 3(0) \implies 2x=12 \implies x=6\). So, one point is C(6, 0).} \\ \bullet & \text{If \(y = -2\), then \(2x = 12 + 3(-2) \implies 2x=6 \implies x=3\). So, another point is D(3, -2).} \\ \end{array}\]

Step 3: Graphing and Solving:
To draw the graph: \begin{enumerate} \item Plot the points A(6, 0) and B(0, 2) on a coordinate plane and draw a straight line passing through them. This line represents the equation \(x + 3y - 6 = 0\). \item Plot the points C(6, 0) and D(3, -2) on the same coordinate plane and draw a straight line passing through them. This line represents the equation \(2x - 3y - 12 = 0\). \item Observe the point where the two lines intersect. Both lines pass through the point (6, 0). \end{enumerate}

Step 4: Final Answer:
The point of intersection of the two lines is (6, 0). Therefore, the solution to the pair of linear equations is \(x = 6\) and \(y = 0\).