Question

Solve the linear equations \(3x + y = 14\) and \(y = 2\) graphically.

Hint:

When solving graphically, choose \(x\) values that result in integer \(y\) values to make plotting easier and more accurate on grid paper.

Answer 1

Step 1: Understanding the Concept:
Graphical solution involves plotting both lines on a Cartesian plane and finding their point of intersection.
Step 2: Key Formula or Approach:
1. Find at least two points for the line \(3x + y = 14\).
2. Plot the line \(y = 2\) (a horizontal line passing through \(y = 2\)).
Step 3: Detailed Explanation:
For \(3x + y = 14\):
- If \(x = 4\), then \(y = 14 - 3(4) = 14 - 12 = 2\). Point is \((4, 2)\).
- If \(x = 0\), then \(y = 14\). Point is \((0, 14)\).
- If \(x = 2\), then \(y = 14 - 3(2) = 14 - 6 = 8\). Point is \((2, 8)\).
Plotting these points and drawing the line gives a straight line.
For \(y = 2\):
- This is a line parallel to the \(x\)-axis passing through 2 on the \(y\)-axis.
Intersection:
Observing the graph, both lines intersect at the point where \(y = 2\). Substituting \(y = 2\) into the first equation:
\[ 3x + 2 = 14 \Rightarrow 3x = 12 \Rightarrow x = 4 \]
The point of intersection is \((4, 2)\).
Step 4: Final Answer:
The solution of the equations is \(x = 4, y = 2\).