What does it mean to "Solve an Equation"?

We often talk about "solving an equation". Unfortunately, we do not always mean the same thing when we say this. In general, if we have an equation that has just one variable, such as x, then "solving the equation" means finding the set of all values that can be substituted for the one variable to produce a valid equation. Thus, to solve

4x – 5 + 7x = 9x + 11 we simplify the equation until we obtain x = 3 which tells us that the set of values that can be substituted for x in the original problem to give a true statement is exactly the set {3}. Likewise, to solve |x – 5| = 13 means finding that both elements of the solution set {18, –8} can be substituted for x in the original equation to give a true statement. In the same way, for the problem x⁴ – 27x² + 14x + 120 = 0 can be solved to produce the solution set {4, 3, –2, –5} In all of these problems we have a single unknown variable. We solve the problem by finding the set of values that can be substituted for the variable to produce a true statement.

If there are two or more variables in an equation, then we can solve the equation "for a variable". We do this by isolating the particular variable on one side of the equation, allowing all other constants and expressions to stand on the other side of the equation. For example, we can solve

5x – 6y = 30 for either x or y. If we solve for x, then we isolate x on one side and we have x = (6y + 30) / 5 However, we could solve for y by isolating the y to get y = (–5x + 30)/(–6) Note that we could give that result in a slightly different form as y = (5x – 30)/(6) Either version is correct, but the second looks nicer in that it does not have the negative value in the denominator.

Notice that the equation

5x – 6y = 30 has a solution set. It is the set of points (x,y) that makes the equation true. As we will see later, and as you probably know now, there are an infinite number of elements in this solution set. It contains elements such as (6,0), (0,–5), (12,5), and (18,10). In also has elements (3,–2.5) and (20,11.6666666...). If we graph the solution set on the coordinate plane, the solution set for 5x – 6y = 30 is a straight line representing an infinite set of ordered pairs that form the solution set of the equation. A graph of that solution set is given as

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Again, these are the ordered pairs that can have their coordinates substituted into the original equation for x and y, respectively, to produce a true statement.

Although the equation

5x – 6y = 30 has an infinite solution set, there are two particular values that generally interest us. These are called the y–intercept and the x–intercept, the points on the graphed line where that line crosses the y–axis and the x–axis, respectively. Remember that all points on the y–axis have an abscissa equal to 0. Similarly, all points on the x–axis have an ordinate that is equal to 0. We can find the y–intercept for the equation 5x – 6y = 30 by setting the value of x to be 0, since we know that the abscissa for that point will be 0. That gives us 5(0) – 6y = 30
^–6y = 30
y = 30 / ^–6
y =^–5 which means that the y-intercept is the point (0,^–5).

In the same way, we can find the x–intercept, the point on the graph where the ordinate is zero, by setting y to be 0, and then solving the resulting equation (for x). In that case we have

5x – 6(0) = 30
5x = 30
x = 30 / 5
y = 6 which means that the x–intercept is the point (6,0).