If $y = \frac{{2x - 1}}{{x + 3}},$ find x when y = 1A. 4B. -4C. 3/2D. 4/3

Question

Prepp Team · Accepted Answer

Solving Algebraic Equations: Finding x when y is Given

The question asks us to find the value of x given the equation relating y and x, and a specific value for y. The given equation is:

$y = \frac{{2x - 1}}{{x + 3}}$

We are given that $y = 1$. Our task is to substitute this value into the equation and then solve for x.

Step-by-Step Solution to Find x

Here is how we can solve for x:

Substitute the given value of y into the equation.
Simplify the resulting equation.
Isolate the variable x on one side of the equation.
Solve for x.

Substituting the Value of y

We are given $y = 1$. Substitute 1 for y in the equation:

$1 = \frac{{2x - 1}}{{x + 3}}$

Solving the Equation for x

Now we need to solve the equation $1 = \frac{{2x - 1}}{{x + 3}}$ for x.

To eliminate the denominator, multiply both sides of the equation by $(x + 3)$, assuming $x \neq -3$:

$1 \times (x + 3) = \frac{{2x - 1}}{{x + 3}} \times (x + 3)$

This simplifies to:

$x + 3 = 2x - 1$

Now, we need to rearrange the equation to bring all terms involving x to one side and constant terms to the other side.

Subtract x from both sides of the equation:

$x + 3 - x = 2x - 1 - x$

$3 = x - 1$

Now, add 1 to both sides of the equation to isolate x:

$3 + 1 = x - 1 + 1$

$4 = x$

So, the value of x is 4 when y is 1.

Comparing the Result with Options

The value we found for x is 4. Let's check the given options:

A. 4
B. -4
C. 3/2
D. 4/3

Our calculated value, $x = 4$, matches option A.

Verification (Optional)

We can verify our answer by plugging $x = 4$ back into the original equation:

$y = \frac{{2(4) - 1}}{{4 + 3}}$

$y = \frac{{8 - 1}}{{7}}$

$y = \frac{{7}}{{7}}$

$y = 1$

This matches the given value of y, confirming that $x = 4$ is the correct solution.

Step	Equation	Action
1	$y = \frac{{2x - 1}}{{x + 3}}$	Original Equation
2	$1 = \frac{{2x - 1}}{{x + 3}}$	Substitute y = 1
3	$1 \times (x + 3) = 2x - 1$	Multiply both sides by $(x+3)$
4	$x + 3 = 2x - 1$	Simplify
5	$3 + 1 = 2x - x$	Rearrange terms (add 1, subtract x)
6	$4 = x$	Solve for x

Revision Table: Key Steps in Solving Equations

Concept	Description	Example Application
Substitution	Replacing a variable with a given value or expression.	Substituting y = 1 into the equation.
Rearranging Formulas	Manipulating an equation to isolate a specific variable.	Moving x terms to one side, constants to the other.
Solving Linear Equations	Finding the value of the unknown variable in an equation where the highest power of the variable is 1.	Solving $x + 3 = 2x - 1$ for x.

Additional Information: Working with Algebraic Equations

When working with algebraic equations, especially those involving fractions, a common first step is to clear the denominators. This is done by multiplying every term in the equation by the least common multiple (LCM) of all denominators. In this specific case, there was only one denominator, $(x + 3)$, so multiplying both sides by $(x + 3)$ was the appropriate step.

After clearing denominators, the equation usually simplifies into a linear equation (like $x + 3 = 2x - 1$) or a quadratic equation, depending on the original expression. Linear equations are solved by collecting all terms with the variable on one side and all constant terms on the other side, and then dividing by the coefficient of the variable.

It's always a good practice, if time permits during an exam, to verify your solution by substituting the calculated value of the variable back into the original equation to ensure it holds true. This helps catch potential errors in calculation or manipulation.

Prepp Team · Answer

C

Prepp Team · Answer

D

Prepp Team · Answer

B