Make use of the space provided to show all working out. QUESTION 1 Solve the following simultaneous equations using graphical method. (Use a grid paper to draw your graph) \[ \begin{array}{l}\text { (6 marks) } \\ \qquad y=4 x-5 \\ \qquad y=-2 x+7\end{array} \]

To solve the simultaneous equations using the graphical method, we will follow these steps: 1. **Identify the equations**: - Equation 1: \( y = 4x - 5 \) - Equation 2: \( y = -2x + 7 \) 2. **Plot the equations on a graph**: - We will find at least two points for each equation to plot them accurately. ### Step 1: Plotting the first equation \( y = 4x - 5 \) To find points for this equation, we can choose values for \( x \) and calculate the corresponding \( y \): - For \( x = 0 \): \[ y = 4(0) - 5 = -5 \quad \Rightarrow \quad (0, -5) \] - For \( x = 1 \): \[ y = 4(1) - 5 = -1 \quad \Rightarrow \quad (1, -1) \] - For \( x = 2 \): \[ y = 4(2) - 5 = 3 \quad \Rightarrow \quad (2, 3) \] ### Step 2: Plotting the second equation \( y = -2x + 7 \) Similarly, we can find points for this equation: - For \( x = 0 \): \[ y = -2(0) + 7 = 7 \quad \Rightarrow \quad (0, 7) \] - For \( x = 1 \): \[ y = -2(1) + 7 = 5 \quad \Rightarrow \quad (1, 5) \] - For \( x = 2 \): \[ y = -2(2) + 7 = 3 \quad \Rightarrow \quad (2, 3) \] ### Step 3: Plotting the points on a graph Now, we can plot the points on a grid paper: - For the first equation \( y = 4x - 5 \): - Points: \( (0, -5) \), \( (1, -1) \), \( (2, 3) \) - For the second equation \( y = -2x + 7 \): - Points: \( (0, 7) \), \( (1, 5) \), \( (2, 3) \) ### Step 4: Drawing the lines 1. Draw a straight line through the points of the first equation. 2. Draw a straight line through the points of the second equation. ### Step 5: Finding the intersection point The solution to the simultaneous equations is the point where the two lines intersect. From our calculations, we can see that both equations yield the point \( (2, 3) \). ### Conclusion The solution to the simultaneous equations is: \[ \boxed{(2, 3)} \] This point represents the values of \( x \) and \( y \) that satisfy both equations.