Solving for

When solving for $x – y$ in a system of equations, we typically deal with two linear equations involving two variables, $x$ and $y$. Let’s go through the process step-by-step with examples to make it clear.

Step-by-Step Guide

  1. Write Down the Equations

    Consider the following system of linear equations:

    1. $2x + 3y = 12$
    2. $4x – y = 5$

  1. Align the Equations

    To make our calculations easier, let’s write the equations one below the other:

    1. $2x + 3y = 12$
    2. $4x – y = 5$

  1. Eliminate One Variable

    To find $x – y$, we need to eliminate one of the variables. This can be done using the method of elimination or substitution. Let’s use elimination here.

    First, we need to make the coefficients of $y$ in both equations equal. To do this, we can multiply the second equation by 3:

    1. $2x + 3y = 12$
    2. $3(4x – y) = 3(5)$

    This simplifies to:

    1. $2x + 3y = 12$
    2. $12x – 3y = 15$

  1. Add or Subtract the Equations

    Next, we add the two equations to eliminate $y$:

    $(2x + 3y) + (12x – 3y) = 12 + 15$

    This simplifies to:

    $14x = 27$

  1. Solve for $x$

    Now, we can solve for $x$:

    $x = frac{27}{14}$

  1. Substitute $x$ Back into One of the Original Equations

    We substitute $x = frac{27}{14}$ back into one of the original equations to find $y$. Let’s use the first equation:

    $2bigg(frac{27}{14}bigg) + 3y = 12$

    Simplify and solve for $y$:

    $frac{54}{14} + 3y = 12$

    $3.857 + 3y = 12$

    $3y = 12 – 3.857$

    $3y = 8.143$

    $y = frac{8.143}{3}$

    $y = 2.714$

  1. Calculate $x – y$

    Now that we have $x$ and $y$, we can find $x – y$:

    $x – y = frac{27}{14} – 2.714$

    $x – y = 1.929 – 2.714$

    $x – y = -0.785$

    So, the value of $x – y$ is approximately $-0.785$

Another Example

Let’s consider another system of equations:

  1. $3x + 4y = 20$
  2. $5x – 2y = 10$

  1. Write Down the Equations

    1. $3x + 4y = 20$
    2. $5x – 2y = 10$

  1. Align the Equations

    1. $3x + 4y = 20$
    2. $5x – 2y = 10$

  1. Eliminate One Variable

    To eliminate $y$, we can multiply the second equation by 2:

    1. $3x + 4y = 20$
    2. $2(5x – 2y) = 2(10)$

    This simplifies to:

    1. $3x + 4y = 20$
    2. $10x – 4y = 20$

  1. Add or Subtract the Equations

    Adding the two equations to eliminate $y$:

    $(3x + 4y) + (10x – 4y) = 20 + 20$

    This simplifies to:

    $13x = 40$

  1. Solve for $x$

    $x = frac{40}{13}$

  1. Substitute $x$ Back into One of the Original Equations

    Substitute $x = frac{40}{13}$ back into the first equation:

    $3bigg(frac{40}{13}bigg) + 4y = 20$

    Simplify and solve for $y$:

    $frac{120}{13} + 4y = 20$

    $9.231 + 4y = 20$

    $4y = 20 – 9.231$

    $4y = 10.769$

    $y = frac{10.769}{4}$

    $y = 2.692$

  1. Calculate $x – y$

    $x – y = frac{40}{13} – 2.692$

    $x – y = 3.077 – 2.692$

    $x – y = 0.385$

    So, the value of $x – y$ is approximately $0.385$

Conclusion

Solving for $x – y$ involves a systematic approach of eliminating one variable, solving for the other, and then substituting back. This method ensures that we find accurate values for both $x$ and $y$, and subsequently $x – y$. Practice with different sets of equations can help you become more comfortable with these steps.

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