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How do you solve $2x + 3y = 6$ and $x + 2y = 5$?

Last updated date: 14th Jun 2024
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Hint: Use the method of elimination to solve this problem. To solve these types of equations, the first step you need to do is to eliminate one of the variables out. To do this, you must multiply each equation with a constant such that the resultant equation when added or subtracted gives only one equation in its equation. Here, we will multiply the first equation by 1 and the second one by 2. After that we will add those two new equations which will eliminate the variable x and give a value for y. Now put that value of y in equation 1 to get the required value of x.

Complete step-by-step solution:
The two fractions we have are
$2x + 3y = 6..........(1)$
$x + 2y = 5..........(2)$
Now, we know that to solve these types of problems, we need to eliminate one of the variables first.But how to do it?
You will need to multiply the first equation by and constant ‘a’ and the second equation ‘b’ such that any one of the unknown variables' coefficients becomes 0.
So, following the above steps, lets multiple equation (1) by 1 and equation (2) by 2
After doing this step, we get the modified equations as:-
$2x + 3y = 6..........(3)$
$2x + 4y = 10..........(4)$
Now, subtract equation (4) from equation (3)
We get,
(2x + 4y) - (2x + 3y) = 10 - 6 \\
\Rightarrow 2x + 4y - 2x - 3y = 4 \\
\Rightarrow y = 4 \\
Therefore, completing this step gave us the value of y as 4
Now putting the value of y in equation (1) we will get
2x + 3 \times 4 = 6 \\
\Rightarrow 2x + 12 = 6 \\
\Rightarrow 2x = - 6 \\
\Rightarrow x = \dfrac{{ - 6}}{2} \\
\Rightarrow x = - 3 \\
Hence, we got the value of x as -3.
So, the solution of the equations $2x + 3y = 6$ and $x + 2y = 5$is $x = - 3$ and $y = 4$

Note: The aim should always be to eliminate only one of the two variables. So, always multiply with such a constant so that you can easily eliminate one of the variables. Also, it is not necessary that you must add the two modified equations. You can also subtract them to get the answer. Use your operator, but make sure you eliminate one variable.