Exercise 5 Page 247

Let's discuss the different ways of solving a system of linear equations.

Addition/Subtraction

If the coefficients of a variable are equal to each other in both equations, we can add or subtract the equations. For example, In this system, the $x$ variable has the same coefficient in both equations but with opposite signs, $3x$ and $\text{-}3x.$ Therefore, we can add these two equations. Then we will obtain an equation with one unknown: This equation gives the value for $y$ as $\frac{9}{7}$. Using this solution in one of the original equations, we can find the corresponding $x\text{-}$value.

Multiplication

If the coefficients of the same variables are not the same, we should first multiply one or both of them by a number so that we obtain their least common multiple. For example, In this system, the variables have different coefficients. Therefore, we need to find the least common multiple of at least one of them. Let's multiply the first equation by $3$ and the second by $2.$ Now, since the $x$ variable has the same coefficient in both of the equations, we can subtract one from the other. Then we have: This equation gives the value for $y$ as $2.$ Using this solution in one of the original equations, we can find the corresponding $x\text{-}$value.