Number of Solutions/Intersections to Linear Equations

When you are asked something regarding the number of solutions to a linear equation or the number of times two lines intersect, it's going to come down to the slopes and y-intercepts of the lines.

Sometimes these types of problems will be asked in the context of a system of linear equations, and sometimes they will be asked in the context of a single linear equation. For example you might be given two functions, f(x) and g(x), where f(x) = 2x + 1 and g(x) = 2x - 1, and then be asked how many times the two lines intersect. Alternatively, sometimes you might be given a single linear equation, and then be asked how many solutions it has. For example, you might be asked how many solutions the equation 2x + 1 = 2x - 1 has. Both of these can be approached similarly - in the latter we can think of the left side of the equation as a function representing a line, and the right side of the equation as a function representing another line. The number of solutions to the equation is then the number of times the two lines intersect.

When a system of linear equations has no solutions, this means the lines never intersect, and are therefore parallel. This means that the lines have the same slope but different y-intercepts. So, for example, if we're told that the equation 3x+5 = kx+10, where k represents some constant, has no solutions, we can conclude the lines represented by 3x+5 and kx+10 are parallel. Therefore, the slopes of the two lines are the same, meaning that k must be equal to 3.

Any time two lines are not parallel, the will eventuall intersect somewhere at a single point. This means that the lines have different slopes. To find what that point is, we can use a couple of different methods, which we'll cover in the next section. The most typical methods of solving a system of linear equations is by using substitution or elimination.

When a system of linear equations has infinitely many solutions, this means the lines are the same line. This means that the lines have the same slope and the same y-intercept. So, for example, if we're told that the equation 3x + 5 = ax + b, where a and b represent some constants, has infinitely many solutions, we can conclude the lines represented by 3x+5 and ax+b are the same line. Therefore, the slopes of the two lines are the same, meaning that a must be equal to 3, and the y-intercepts must be the same, meaning that b must be equal to 5.