Let's break down the solution to the simultaneous equations using the graph and the given instructions.

Part (a): Graphical Solution

We are asked to use the graph to find the solution to the two simultaneous equations:

$y = 2x + 1 \quad \text{(purple line)}$ $y = -x + 7 \quad \text{(green line)}$

From the graph, these two lines intersect at a point. The coordinates of the intersection give the solution to the simultaneous equations.

Looking at the graph:

• The lines intersect at the point $(2, 5)$.
• Therefore, the solution to the simultaneous equations is $x = 2$ and $y = 5$.

Part (b): Substituting $x = 2$ into the equations

1. Substitute $x = 2$ into the first equation $y = 2x + 1$: $y = 2(2) + 1 = 4 + 1 = 5$ So, $y = 5$ for the first equation.

2. Substitute $x = 2$ into the second equation $y = -x + 7$: $y = -(2) + 7 = -2 + 7 = 5$ So, $y = 5$ for the second equation as well.

What do you notice?

• i) The value of $y$ is the same for both equations, $y = 5$, when $x = 2$.
• ii) The value of $y = 5$ matches the value we found at the intersection point in part (a).

Summary:

• The solution from the graph is $(2, 5)$.
• Substituting $x = 2$ into both equations confirms that $y = 5$, which matches the graphical solution.

Would you like any further details or clarifications on this problem?

Here are 5 related questions to expand on this topic:

1. How do you solve simultaneous equations algebraically without using a graph?
2. What does it mean when two lines on a graph are parallel in terms of solutions to simultaneous equations?
3. Can you have simultaneous equations with no solution? How would that look on a graph?
4. How does substitution differ from elimination when solving simultaneous equations?
5. How can you verify the solution of a system of equations using a table of values?

Tip: When solving simultaneous equations graphically, always check the point of intersection to ensure it satisfies both equations.