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How do you solve $y = \frac{1}{x}$ $y= 1/x$ and $x + 5 y = 6$ $x+5y=6$ using substitution?

1 Answer

Jul 27, 2016

$(x, y) = (1, 1) or (5, \frac{1}{5})$ $(x,y)=color(green)(""(1,1)) " or " color(green)(""(5,1/5))$

Explanation:

Given
[1] $XXX y = \frac{1}{x}$ $color(white)("XXX")y=1/x$
[2] $XXX x + 5 y = 6$ $color(white)("XXX")x+5y=6$

Using [1] we can substitute $\frac{1}{x}$ $1/x$ for $y$ $y$ in [2]
[3] $XXX x + \frac{5}{x} = 6$ $color(white)("XXX")x+5/x=6$

Multiplying both sides by $x$ $x$
[4] $XXX x^{2} + 5 = 6 x$ $color(white)("XXX")x^2+5=6x$

Rearranging into standard quadratic form:
[5] $XXX x^{2} - 6 x + 5 = 0$ $color(white)("XXX")x^2-6x+5=0$

Factoring
[6] $XXX (x - 1) (x - 5) = 0$ $color(white)("XXX")(x-1)(x-5)=0$

Allowing the two solutions:
${:(x=1,color(white)("XX"),x=5),(rarr y=1/1=1,,rarr y=1/5):}$

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