Exercise 19 Page 411

A system of linear equations with two linear equations can have zero, one or infinitely many solutions. It is possible to find the number of solutions of a system by solving the equations algebraically. To do so, we need to check the result we get by using either the Substitution Method or the Elimination Method.

1. One solution: The result is a solution for one variable.
2. No solution: The result is a false statement.
3. Infinitely many solutions: The result is an identity.

We are given that the result we get is x=8 when we add two linear equations in a system. We can find the value of y by substituting x=8 into any of the equations. Hence, the system has one solution and it is the point (8,y).

Extra

Example

To make sense of the explanation, let's consider an example system that will result in x=8 when we add the equations in the system. 2x-y= 14 & (I) y-x= - 6 & (II) We will add these equations. 2x-y+ y-x= 14+( - 6) ⇓ x=8 We can now substitute x=8 into one of the equations. Let's use Equation (II).

y-x=- 6

Substitute

x= 8

y- 8=- 6

AddEqn

LHS+8=RHS+8

y=2

We found that the point (8,2) satisfies both equations in the system. It means that the lines that represent the equations intersect at (8,2). Let's look at the graph of the system to visualize it.