Solve the simultaneous equations:
2x + 8y - 3z = -1
2x - 3y + 2z = 2
2x + y + z = 5
The graphs of the functions: f : x ↦ |x - 3| and g : x ↦ 2 are shown in the diagram - Leaving Cert Mathematics - Question 4 - 2018
Question 4
Solve the simultaneous equations:
2x + 8y - 3z = -1
2x - 3y + 2z = 2
2x + y + z = 5
The graphs of the functions: f : x ↦ |x - 3| and g : x ↦ 2 are shown in the dia... show full transcript
Worked Solution & Example Answer:Solve the simultaneous equations:
2x + 8y - 3z = -1
2x - 3y + 2z = 2
2x + y + z = 5
The graphs of the functions: f : x ↦ |x - 3| and g : x ↦ 2 are shown in the diagram - Leaving Cert Mathematics - Question 4 - 2018
Step 1
Solve the simultaneous equations:
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Answer
To solve the simultaneous equations, we will utilize substitution and elimination methods. Starting with the equations:
2x+8y−3z=−1
2x−3y+2z=2
2x+y+z=5
We can manipulate these equations:
First, subtract the second equation from the first:
2x+8y−3z−(2x−3y+2z)=−1−2
Which simplifies to:
11y−5z=−3
Now, we can subtract the third equation from the first:
2x+8y−3z−(2x+y+z)=−1−5
This results in:
7y−4z=−6
Now we have two new equations:
11y−5z=−3
7y−4z=−6
We can now solve for one variable in terms of the others. From the first equation, multiplying by 7:
77y−35z=−21
From the second equation, multiplying by 11:
77y−44z=−66
Subtracting these gets us:
9z=45
Hence, we find:
z=5
Substituting z=5 back into one of the original equations to find y yields:
7y=20exthencey=2
Finally, substituting both y and z into any of the original equations to find x:
2x+8(2)−3(5)=−1
This gives:
2x+16−15=−1
So:
2x=−2exthencex=−1
Thus, the solution is:
x=−1,y=2,z=5
Check the results against original equations to confirm correctness.
Step 2
Find the co-ordinates of the points A, B, C and D.
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Answer
To find the coordinates of the points A, B, C, and D:
From the diagram, identify point D where the graph of g(x) intersects the y-axis. At this point, x=0:
g(0)=2oD=(0,2)
Point C is where f(x) intersects the x-axis. Since f(x)=∣x−3∣, this occurs when:
∣x−3∣=0ox=3oC=(3,0)
Points A and B occur where f(x)=g(x), solving:
∣x−3∣=2
This results in two equations:
x−3=2ox=5
x−3=−2ox=1
Hence, A = (1, 2) and B = (5, 2).
The coordinates are:
A = (1, 2)
B = (5, 2)
C = (3, 0)
D = (0, 2)
Step 3
Hence, or otherwise, solve the inequality |x - 3| < 2.
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Answer
The inequality ∣x−3∣<2 translates to:
x−3<2 and
x−3>−2
This leads to:
x<5
x>1
Thus, the solution set for this inequality is:
1<x<5
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