Chapter 51: Problem 2

Rearrange the following formulae: (a) If \(y=a x+b\), find \(b\) (b) If \(y=a x+b\), find \(x\) (c) If \(x=y^{3}\), find \(y\) (d) If \(x=3^{y}\), find \(y\) (e) If \(x=(1-y)\left(z^{p}+3\right)\), find \(z\) (f) If \(x=(y-z)^{1 / n} / p q\), find \(n\)

Short Answer

Expert verified

The solutions are: (a) \(b = y - ax\), (b) \(x = (y-b)/a\), (c) \(y = \sqrt[3]{x}\), (d) \(y = \ln(x)/\ln(3)\), (e) \(z = \sqrt[p]{\frac{x - 3 + 3y}{1-y}}\), (f) \(n = \log_{y-z}(pqx)\).

Step by step solution

Finding \(b\)

You can isolate \(b\) in the equation \(y=ax+b\) by subtracting \(ax\) from both sides of the equation. This will give you \(b\) on one side and the other terms on the other. Thus, \(b = y - ax\).

Finding \(x\)

You can isolate \(x\) in the equation \(y=ax+b\) by subtracting \(b\) from both sides and then dividing by \(a\). So, \(x = (y-b)/a\).

Finding \(y\)

To find \(y\) in the equation \(x=y^3\), take the cube root of both sides. Thus, \(y = \sqrt[3]{x}\).

Finding \(y\)

To isolate \(y\) in the equation \(x=3^y\), you should take the natural logarithm (ln) of both sides and then divide both sides of the equation by ln(3) to get \(y = \ln(x)/\ln(3)\).

Finding \(z\)

To isolate \(z\) in the equation \(x=(1-y)(z^p+3)\), first distribute the expression to get \(x = z^p - yz^p + 3 - 3y\). Then isolate the terms containing \(z\) by subtracting \(3 - 3y\) from both sides to get \(z^p - yz^p = x - 3 + 3y\). Factor out a \(z^p\), then divide both sides by \((1-y)\), yielding \(z = \sqrt[p]{\frac{x - 3 + 3y}{1-y}}\).

Finding \(n\)

To isolate \(n\) in the equation \(x=(y-z)^{1/n}/pq\), multiply both sides by \(pq\) to get \(pqx = (y-z)^{1/n}\). Then raise both sides to the power of \(n\) to yield \(n = \log_{y-z}(pqx)\).

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Rearrange the following formulae: (a) If \(y=a x+b\), find \(b\) (b) If \(y=a x+b\), find \(x\) (c) If \(x=y^{3}\), find \(y\) (d) If \(x=3^{y}\), find \(y\) (e) If \(x=(1-y)\left(z^{p}+3\right)\), find \(z\) (f) If \(x=(y-z)^{1 / n} / p q\), find \(n\)

Short Answer

Step by step solution

Finding \(b\)

Finding \(x\)

Finding \(y\)

Finding \(y\)

Finding \(z\)

Finding \(n\)

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