Question

How do you solve the system $2x - y = 8$ and $x + 2y = 9$

Answer 1

Hint: The given system is linear equations in two variables (since the two variables are in first power). We can solve the equations by using substitution method where we make the coefficients of one variable equal in both the equations (by multiplying it with a certain factor) and subtract/add the equations and find the value of one variable and then substitute it in any one of the equations to find the value of the other.

Complete step-by-step answer:
Let’s name the equations 1 and 2 for easy reference.
$2x - y = 8.....(1)$
$x + 2y = 9.....(2)$
We will select the variable x. The coefficients of x in both the equations are 2 and 1. We can simply multiply the second equation with 2 and then subtract the two equations to find the value of y.
If we select the variable y, the coefficients of y in both the equations are -1 and 2. We can multiply the first equation with 2 and then add the two equations to find the value of x.
After multiplying the second equation with 2, we get:
$2x - y = 8$
$2x + 4y = 18$
We can subtract equation 1 from equation 2. After subtracting we get:
$(2x + 4y) - (2x - y) = 18 - 8$
$5y = 10$
$y = 2$
We can use the value of y to find the value of x. Substituting the value of y in the first equation, we get:
$2x - 2 = 8$
$2x = 10$
$x = 5$
We can verify the values by substituting them in the second equation. After substituting, we get:
$5 + 2(2) = 9$
Since both sides are equal, our values are correct.
$(x,y) = (5,2)$

Note: When we have a system of linear equations of two variables, we can solve them by using various methods like elimination, substitution, graphical methods etc. We need to select the most suitable method which involves least calculation and complexity. When we subtract and add the equations, we need to be careful with negative and positive signs because a simple mistake can give a wrong answer.