15. Can x, y be found to satisfy the following equations simultaneously?
3/x-2y=1
4/x+5y=2
9x - 4y + 23 = 0
If so, find them.
Answers
Answer:
Given :
A body has a displacement (2, 4, -6) to (6, -4, 4) under a constant force of \sf{2\hat{i}+3\hat{j}-\hat{k}}2
i
^
+3
j
^
−
k
^
To find :
Work done
Knowledge required :
Dot product of two different orthogonal vectors is equals to 0 ; \sf{\hat{i}\;.\;\hat{j}=\hat{j}\;.\;\hat{k}=\hat{k}\;.\;\hat{i}=0}
i
^
.
j
^
=
j
^
.
k
^
=
k
^
.
i
^
=0
and Dot product of an orthogonal vector with itself is equals to 1 ; \sf{\hat{i}\;.\;\hat{i}=\hat{j}\;.\;\hat{j}=\hat{k}\;.\;\hat{k}=1}
i
^
.
i
^
=
j
^
.
j
^
=
k
^
.
k
^
=1
Formulae to calculate work done
\star\;\;\boxed{\sf{W=\vec{F}\;.\;\vec{d}}}⋆
W=
F
.
d
[ Work done is given by the dot product of force and displacement vector ; where F is Force and d is displacement ]
Solution :
Given, position vectors of the body
\sf{\vec{r_1}=2\;\hat{i}+4\;\hat{j}-6\;\hat{k}}
r
1
=2
i
^
+4
j
^
−6
k
^
and
\sf{\vec{r_2}=6\;\hat{i}-4\;\hat{j}+4\;\hat{k}}
r
2
=6
i
^
−4
j
^
+4
k
^
Calculating displacement of body
Displacement is known as the change in position vector.
so,
\implies\sf{\vec{d}=\vec{r_2}-\vec{r_1}}⟹
d
=
r
2
−
r
1
\implies\sf{\vec{d}=(6\;\hat{i}-4\;\hat{j}+4\;\hat{k})-(2\;\hat{i}+4\;\hat{j}-6\;\hat{k})}⟹
d
=(6
i
^
−4
j
^
+4
k
^
)−(2
i
^
+4
j
^
−6
k
^
)
\implies\sf{\vec{d}=6\;\hat{i}-4\;\hat{j}+4\;\hat{k}-2\;\hat{i}-4\;\hat{j}+6\;\hat{k}}⟹
d
=6
i
^
−4
j
^
+4
k
^
−2
i
^
−4
j
^
+6
k
^
\underline{\implies\sf{\vec{d}=4\;\hat{i}-8\;\hat{j}+10\;\hat{k}}}
⟹
d
=4
i
^
−8
j
^
+10
k
^
Calculating Work done by the body
Given,
\sf{\vec{F}=2\;\hat{i}+3\;\hat{j}-\;\hat{k}}
F
=2
i
^
+3
j
^
−
k
^
Using formula
\implies\sf{W=\vec{F}\;.\;\vec{d}}⟹W=
F
.
d
\implies\sf{W=(2\;\hat{i}+3\;\hat{j}-\;\hat{k})\;.\;(4\;\hat{i}-8\;\hat{j}+10\;\hat{k})}⟹W=(2
i
^
+3
j
^
−
k
^
).(4
i
^
−8
j
^
+10
k
^
)
\implies\sf{W=8-24-10}⟹W=8−24−10
\overbrace{\underbrace{\boxed{\implies\sf{W=-26\;\;Units}}}}
⟹W=−26Units
Therefore,
Magnitude of Work done will be equal to 26 Units .
Answer:
How to Solve for Both X & Y
Rewrite the linear equations in standard form Ax + By = 0 by combining like terms and adding or subtracting terms from both sides of the equation. ...
Write one of the equations directly underneath one another so the x and y variable.
The Elimination Method
Step 1: Multiply each equation by a suitable number so that the two equations have the same leading coefficient. ...
Step 2: Subtract the second equation from the first.
Step 3: Solve this new equation for y.
Step 4: Substitute y = 2 into either Equation 1 or Equation 2 above and solve for x.
hope it is helpful
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