Question

4 (a) make two copies of table 16.4.
table 16.4
\

$$\begin{tabular}{|c|c|c|c|} \\hline x & -4 & 0 & +2 \\\\ \\hline y & & & \\\\ \\hline \\end{tabular}$$

(b) complete the tables for \\(y = 1 - 2x\\) and \\(y = x + 7\\) and draw the graphs of these equations within the same axes.
(c) use the graphs to solve the equations simultaneously.
5 solve the following pairs of simultaneous equations graphically.
(a) \\(y = 3x\\)
\\(y = x + 1\\)
(b) \\(3x - y = 12\\)
\\(4x + 2y = 1\\)
(hint: in (b), first make \\(y\\) the subject of each

Explanation:

Complete the tables

For \(y = 1 - 2x\):

• When \(x = -4\), \(y = 1 - 2(-4) = 9\).
• When \(x = 0\), \(y = 1 - 2(0) = 1\).
• When \(x = 2\), \(y = 1 - 2(2) = -3\).

For \(y = x + 7\):

• When \(x = -4\), \(y = -4 + 7 = 3\).
• When \(x = 0\), \(y = 0 + 7 = 7\).
• When \(x = 2\), \(y = 2 + 7 = 9\).

Find the intersection

Set the two equations equal to find the simultaneous solution:

$$ 1 - 2x = x + 7 $$

$$ -3x = 6 \implies x = -2 $$

Substitute \(x = -2\) back:

$$ y = -2 + 7 = 5 $$

The intersection point is \((-2, 5)\).

Solve Question 5(a) graphically

Equations:

1. \(y = 3x\)
2. \(y = x + 1\)

Find the intersection:

$$ 3x = x + 1 \implies 2x = 1 \implies x = 0.5 $$

$$ y = 3(0.5) = 1.5 $$

The solution is \((0.5, 1.5)\).

Solve Question 5(b) graphically

Equations:

1. \(3x - y = 12 \implies y = 3x - 12\)
2. \(4x + 2y = 1 \implies 2y = -4x + 1 \implies y = -2x + 0.5\)

Find the intersection:

$$ 3x - 12 = -2x + 0.5 $$

$$ 5x = 12.5 \implies x = 2.5 $$

$$ y = 3(2.5) - 12 = 7.5 - 12 = -4.5 $$

The solution is \((2.5, -4.5)\).

Answer:

Question 4

(a) & (b) Completed tables:

Table for \(y = 1 - 2x\):

\(x\)	\(-4\)	\(0\)	\(+2\)

Table for \(y = x + 7\):

\(x\)	\(-4\)	\(0\)	\(+2\)

(c) The simultaneous solution from the intersection of the graphs is:

$$x = -2, \quad y = 5$$

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Question 5

(a) The graphical solution to the simultaneous equations is:

$$x = 0.5, \quad y = 1.5$$

(b) The graphical solution to the simultaneous equations is:

$$x = 2.5, \quad y = -4.5$$