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Solve 1/2+1/2x=x^2-7x+10/4x by re-writing the equation as a proportion. Which proportion is equivalent to the original equation?

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zoexoe
Rewriting the equation as a proportion, we have
     (1/2 * 2x/2x) + (1/2x * 2/2) = (x^2 - 7x + 10)/4x
     (2x/4x) + (2/4x) = (x^2 - 7x + 10)/4x

Multiplying both sides of the equation by 4x to clear the denominators:
     2x + 2 = x^2 - 7x + 10
We now have a new equation that is equivalent to the original equation:
     x^2 - 9x + 8 = 0

We can also write the equation into its factored form:
     (x - 8)(x - 1) = 0
Now we have a product of two expressions that is equal to zero, which means any x value that makes either (x - 8) or (x - 1) zero will make their product zero.
     x - 8 = 0  => x = 8
     x - 1 = 0  => x = 1
Therefore, our solutions are x = 8 and x = 1.
abidemiokin

The value of x from the system of equation will be 1 and 8 and the proportion equivalent is x^2 - 9x + 8 =0

Given the expression [tex]\frac{1}{2} +\frac{1}{2x} = \frac{x^2-7x+10}{4x}[/tex]

Multiply the first term by 2x/2x  and second term by 2/2 to have:

[tex]\frac{2x}{4x} +\frac{2}{4x} = \frac{x^2-7x+10}{4x}[/tex]

Multiply through by 4x to have:

[tex]2x + 2 = x^2-7x+10[/tex]

Collect the like terms and factorize

[tex]2x + 2 = x^2-7x+10\\ x^2-7x+10-2x-2=0\\ x^2-9x+8=0\\x^2-8x-x+8=0\\x(x-8)-1(x-8)=0\\(x-1)(x-8)=0\\x=1 \ and \ 8[/tex]

Hence the value of x from the system of equation will be 1 and 8 and the proportion equivalent is x^2 - 9x + 8 =0

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