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Suppose you are solving a system such as 2x + 3y = 3 4x + 6y = 6 Solving this system with substation or elimination leads to 0 = 0. This is a signal that there are an
infinite number of solutions. This does not mean that ANY ordered pairs will solve the system. Only certain combinations of x and y will work. You need a way of finding any of those solutions. There are two ways to do this. Method 1: Start with one of the equations (it does not matter which one) like 2x + 3y = 3. Solve this equation for x: 2x = -3y + 3 and then If you have a value for
y, this gives you a corresponding value for x. For instance, if y = 1, the corresponding x value is x = 0. This gives us one possible solution, (0, 1). If y = -1, then x = 3 giving us (3, -1). In general, we can write all solutions out as Picking any value for y will give you a corresponding value for x which solves the system.
Method 2: What if we were to take 2x + 3y = 3 and solve for y. In this case we would get . If we were to pick values for x, we get corresponding values for x. For instance, if x = 0 we get y = 1 or the ordered pair (0, 1). Notice that this is one of the same ordered pairs as in Method 1. Let’s try another value for x, x = 3. When we put this into we get y = -1. This gives the same ordered pair, (3, -1), as Method 1. In general, we can write out all possible solutions as
Both ways of writing the solution give the same ordered pairs. In Method 1, you pick a value for y and find the corresponding x value. In Method 2, you pick a value for x and find the corresponding y value. Since the values you pick can be anything, this gives the infinite number of ordered pairs that solve the system.
Solving Systems of Linear Equations by Substitution
Graphing is a useful tool for solving systems of equations, but it can sometimes be time-consuming. A quicker way to solve systems is to isolate one variable in one equation, and substitute the resulting expression for that variable in the other equation. Observe:
Example 1: Solve the following system, using substitution:
The easiest variable to isolate is y in the first equation, because it has no coefficient:
y = 13 - 5x
In the second equation, substitute for y its equivalent expression:
3x = 15 - 3(13 - 5x)
Solve the equation:
3x = 15 - 39 + 15x
3x = 15x - 24
-12x = - 24
x = 2
Now substitute this x-value into the "isolation
equation" to find y:
y = 13 - 5x = 13 - 5(2) = 13 - 10 = 3
Thus, the solution to the system is (2, 3). It is useful to check this solution in both equations.
Note: Although we chose y in the first equation in the previous example, isolating any variable in any equation will yield the same solution.
Example 2: Solve the following system, using substitution:
It is easier to work with the second equation, because there is no constant term:
5x = 10y
x = 2y
In the first equation, substitute for x its equivalent expression:
2(2y) + 4y = 36
Solve the equation:
4y + 4y = 36
8y = 36
y = 4.5
Plug this y-value into the isolation
equation to find x:
x = 2y = 2(4.5) = 9
Thus, the solution to the system is (9, 4.5).
Example 3: Solve the following system, using substitution:
It is easiest to isolate x in the second equation, since the x term already stands alone:
x =
x = 7 + 2y
In the first equation, substitute for x its equivalent expression:
2(7 + 2y) - 4y = 12
Solve the equation:
14 + 4y - 4y = 12
14 = 12
Since 14≠12, the system of equations has no solution. It is inconsistent (and independent). The two equations describe two parallel lines.
Example 4: Solve the following system, using substitution:
Either equation can be used to isolate the variable. We will isolate y in the second equation:
2y = 5x + 34
y =
y =
In the first equation, substitute for
y its equivalent expression:
10x = 4(x + 17) - 68
10x = 10x + 68 - 68
10x = 10x
0 = 0
Since 0 = 0 for any value of x, the system of equations has infinite solutions. Every ordered pair (x, y) which satisfies y = x + 17 (the isolation equation) is a solution to the system. The system is dependent (and consistent). The two equations describe the same line--y = x + 17.