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# Solve for x and y: a²x+b²y=c², b²x+a²y=d²

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This is the basic and conceptual question from linear equations in two variables in which we have given two equations a²x+b²y=c², b²x+a²y=d² and we have been asked to find the value of variables x and y from the given equations

Kindly solve the equations

RS Aggarwal, Class 10, chapter 3B, question no 49

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1. Hope it helps you.

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2. Sol:
The given equations are
a
2
x + b2
y = c2
………(i)
b
2
x + a2
y = d2
………(ii)
Multiplying (i) by a2
and (ii) by b2
and subtracting, we get
a
4
x – b
4
x = a2
c
2
– b
2
d
2
⇒x =
𝑎
2𝑐
2−𝑏
2𝑑
2
𝑎4− 𝑏4
Now, multiplying (i) by b2
and (ii) by a2
and subtracting, we get
b
4
y – a
4
y = b2
c
2
– a
2
d
2
⇒y =
𝑏
2𝑐
2−𝑎
2𝑑
2
𝑏4− 𝑎4
Hence, x = 𝑎
2𝑐
2−𝑏
2𝑑
2
𝑎4− 𝑏4
and y = 𝑏
2𝑐
2−𝑎
2𝑑
2
𝑏4− 𝑎4
.

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3. Solve for x and y:
a
2
x + b2
y = c2
, b2
x + a2
y = d2
Sol:
The given equations are
a
2
x + b2
y = c2
………(i)
b
2
x + a2
y = d2
………(ii)
Multiplying (i) by a2
and (ii) by b2
and subtracting, we get
a
4
x – b
4
x = a2
c
2
– b
2
d
2
⇒x =
𝑎
2𝑐
2−𝑏
2𝑑
2
𝑎4− 𝑏4
Now, multiplying (i) by b2
and (ii) by a2
and subtracting, we get
b
4
y – a
4
y = b2
c
2
– a
2
d
2
⇒y =
𝑏
2𝑐
2−𝑎
2𝑑
2
𝑏4− 𝑎4
Hence, x = 𝑎
2𝑐
2−𝑏
2𝑑
2
𝑎4− 𝑏4
and y = 𝑏
2𝑐
2−𝑎
2𝑑
2
𝑏4− 𝑎4
.

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