Question 233·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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The equation $\dfrac{a}{b-c} = \dfrac{2b}{c-a}$ relates the real numbers $a$ , $b$ , and $c$ , where $b \ne c$ and $c \ne a$ . Which equation correctly expresses $c$ in terms of $a$ and $b$ ?

For this kind of rational equation where you must solve for one variable in terms of others, first clear denominators by cross-multiplying (only valid because the denominators are nonzero). Then expand, collect all terms containing the target variable on one side, and factor that variable out. Finally, isolate it by dividing by its coefficient. Pay close attention to sign changes when moving terms across the equals sign, and double-check that you solved for the variable itself (not its reciprocal) before matching to the answer choices.

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First remove the fractions

You have $\dfrac{a}{b-c} = \dfrac{2b}{c-a}$ . What operation lets you get rid of both denominators at once?

Expand and group like terms

After you clear the denominators, distribute on both sides and then move terms so that every term containing $c$ is on one side of the equation.

Factor out the variable you want

Once all $c$ terms are together, factor $c$ out of that side. Then think about what you would divide by to leave $c$ alone.

Check the structure of your result

Your final expression for $c$ should be a single fraction with a sum of squares in the numerator and a sum (not a difference) of $a$ and $b$ in the denominator.

Desmos Guide

Pick simple values for $a$ and $b$

In Desmos, define $a$ and $b$ as simple constants, such as a = 1 and b = 3 (any real numbers are fine, as long as you avoid choices that obviously make denominators zero later, like setting $a + 2b = 0$ when you test formulas).

Use Desmos to solve the original equation for $c$ numerically

Enter the function in terms of $c$ , for example: f(c) = a/(b - c) - 2b/(c - a). Then graph y = f(c) and find the value of $c$ where the graph crosses the $x$ -axis (where $f(c) = 0$ ). That $x$ -coordinate is the numeric value of $c$ that satisfies the original equation for your chosen $a$ and $b$ .

Evaluate each answer choice’s expression for $c$

Create expressions for each option, using your chosen $a$ and $b$ values, such as A = (2b^2 + a^2)/(a + 2b), B = (2b^2 - a^2)/(a - 2b), etc. Compare these numeric values with the $c$ you found from the graph; the option whose value matches that $c$ is the correct rearranged formula.

Step-by-step Explanation

Clear the denominators by cross-multiplying

Start from the given equation:

\frac{a}{b-c} = \frac{2b}{c-a}

Since $b \ne c$ and $c \ne a$ , both denominators are nonzero, so we can cross-multiply:

a(c - a) = 2b(b - c)

Expand both sides

Distribute on each side:

Left side: $a(c - a) = ac - a^2$
Right side: $2b(b - c) = 2b^2 - 2bc$

So the equation becomes:

ac - a^2 = 2b^2 - 2bc

Collect all terms with $c$ on one side

Move the $-2bc$ term to the left and the $-a^2$ term to the right. Add $2bc$ to both sides and add $a^2$ to both sides:

ac - a^2 + 2bc = 2b^2

ac + 2bc = a^2 + 2b^2

Now all the $c$ terms are on the left; the rest are on the right.

Factor out $c$ and solve for $c$

Factor $c$ out of the left-hand side:

c(a + 2b) = a^2 + 2b^2

Now divide both sides by $a + 2b$ (assuming $a + 2b \ne 0$ ):

c = \frac{a^2 + 2b^2}{a + 2b}

This matches choice D) $c = \dfrac{2b^2 + a^2}{a + 2b}$ .

Question 233·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation