QUESTION IMAGE
Question
4 (a) make two copies of table 16.4.
table 16.4
\
(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
<pre_analysis>
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"Graphical Method"
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<reasoning>
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:
- \(y = 3x\)
- \(y = x + 1\)
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:
- \(3x - y = 12 \implies y = 3x - 12\)
- \(4x + 2y = 1 \implies 2y = -4x + 1 \implies y = -2x + 0.5\)
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 \((2.5, -4.5)\).
</reasoning>
<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:
\[x = -2, \quad y = 5\]
---
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:
\[x = 2.5, \quad y = -4.5\]
</answer>
<plot>
{
"elements": [
{
"type": "functiongraph",
"params": [{"js": "1 - 2*x", "latex": "y = 1 - 2x"}, -6, 4],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2,
"name": "y = 1 - 2x",
"withLabel": true
}
},
{
"type": "functiongraph",
"params": [{"js": "x + 7", "latex": "y = x + 7"}, -6, 4],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 2,
"name": "y = x + 7",
"withLabel": true
}
},
{
"type": "point",
"params": [[-2, 5]],
"properties": {
"name": "(-2, 5)",
"size": 4,
"color": "#F2557F",
"withLabel": true
}
}
],
"timestamps": [0.5, 1.0, 1.5]
}
</plot>
<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Algebra",
"Simultaneous Equations"
]
}
</post_analysis>
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],
"current_concepts": [
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</pre_analysis>
<reasoning>
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:
- \(y = 3x\)
- \(y = x + 1\)
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:
- \(3x - y = 12 \implies y = 3x - 12\)
- \(4x + 2y = 1 \implies 2y = -4x + 1 \implies y = -2x + 0.5\)
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 \((2.5, -4.5)\).
</reasoning>
<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:
\[x = -2, \quad y = 5\]
---
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:
\[x = 2.5, \quad y = -4.5\]
</answer>
<plot>
{
"elements": [
{
"type": "functiongraph",
"params": [{"js": "1 - 2*x", "latex": "y = 1 - 2x"}, -6, 4],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2,
"name": "y = 1 - 2x",
"withLabel": true
}
},
{
"type": "functiongraph",
"params": [{"js": "x + 7", "latex": "y = x + 7"}, -6, 4],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 2,
"name": "y = x + 7",
"withLabel": true
}
},
{
"type": "point",
"params": [[-2, 5]],
"properties": {
"name": "(-2, 5)",
"size": 4,
"color": "#F2557F",
"withLabel": true
}
}
],
"timestamps": [0.5, 1.0, 1.5]
}
</plot>
<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Algebra",
"Simultaneous Equations"
]
}
</post_analysis>