How to Find x + y from Given Equations?

Finding the sum of $x$ and $y$ from given equations usually involves solving a system of linear equations. Let’s walk through this process step by step.

Step-by-Step Process

1. Write Down the Equations
   Suppose we have two linear equations:
   1. $a_1x + b_1y = c_1$
   2. $a_2x + b_2y = c_2$

2. Solve One of the Equations for One Variable
   Choose one of the equations and solve for either $x$ or $y$. For instance, solve the first equation for $x$:

   $x = frac{c_1 – b_1y}{a_1}$

3. Substitute the Expression into the Other Equation
   Substitute the expression for $x$ into the second equation:

   $a_2frac{c_1 – b_1y}{a_1} + b_2y = c_2$

4. Solve for y
   Simplify and solve for $y$:

   $frac{a_2c_1 – a_2b_1y}{a_1} + b_2y = c_2$

   Multiply through by $a_1$ to clear the fraction:

   $a_2c_1 – a_2b_1y + a_1b_2y = a_1c_2$

   Combine like terms:

   $a_2c_1 + (a_1b_2 – a_2b_1)y = a_1c_2$

   Solve for $y$:

   $y = frac{a_1c_2 – a_2c_1}{a_1b_2 – a_2b_1}$

5. Substitute Back to Find x
   Now substitute this value of $y$ back into the expression for $x$:

   $x = frac{c_1 – b_1frac{a_1c_2 – a_2c_1}{a_1b_2 – a_2b_1}}{a_1}$

   Simplify to find $x$

6. Add x and y
   Finally, add the values of $x$ and $y$ to find $x + y$:

   $x + y = frac{c_1 – b_1frac{a_1c_2 – a_2c_1}{a_1b_2 – a_2b_1}}{a_1} + frac{a_1c_2 – a_2c_1}{a_1b_2 – a_2b_1}$

Example

Let’s use an example to make this clearer. Suppose we have the equations:

1. $2x + 3y = 6$
2. $4x – y = 5$

1. Write Down the Equations
   We already have them:
   1. $2x + 3y = 6$
   2. $4x – y = 5$

2. Solve One of the Equations for One Variable
   Solve the first equation for $x$:

   $x = frac{6 – 3y}{2}$

3. Substitute the Expression into the Other Equation
   Substitute into the second equation:

   $4frac{6 – 3y}{2} – y = 5$

   Simplify:

   $2(6 – 3y) – y = 5$

   $12 – 6y – y = 5$

   $12 – 7y = 5$

4. Solve for y

   $-7y = -7$

   $y = 1$

5. Substitute Back to Find x
   Substitute $y = 1$ back into the expression for $x$:

   $x = frac{6 – 3(1)}{2}$

   $x = frac{6 – 3}{2}$

   $x = frac{3}{2}$

6. Add x and y

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

   $x + y = frac{3}{2} + frac{2}{2}$

   $x + y = frac{5}{2}$

   So, $x + y = frac{5}{2}$

Conclusion

By following these steps, you can find the sum of $x$ and $y$ from a system of linear equations. Practice with different sets of equations to become more comfortable with this process.

Citations

1. 1. Khan Academy – Systems of Linear Equations
2. 2. Purplemath – Solving Systems of Linear Equations
3. 3. Math is Fun – Solving Simultaneous Equations