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The substitution method is one of several methods used for solving systems of equations.  If, for example, you have a system of two linear equations in x and y, you could solve either equation for x and subst. the resulting expression in y into the other equation.  This eliminates x and allows you to find the value of y.

Example:  Solve this system of linear equations:

3x + 4y = 7
2x  +  y  = -3

Let's solve the 2nd equation for y:  y = -2x - 3.  Now substitute -2x -3 for y in the other equation:     3x + 4(-2x-3) = 3   (Note how y disappears!)

Then 3x -8x - 12 = 3; -5x = 15  => x=-3.

Again we substitute:  Subst. -3 for x in either of the given equations and solve for y.

Write your solution in the form (x,y).

These are examples of the subst. method and its applications.

The substitution method for solving linear systems

A way to solve a linear system algebraically is to use the substitution method. The substitution method functions by substituting the one y-value with the other. We're going to explain this by using an example.

y=2x+4

3x+y=9

We can substitute y in the second equation with the first equation since y = y.

3x+y=9

3x+(2x+4)=9

5x+4=9

5x=5

x=1

This value of x can then be used to find y by substituting 1 with x e.g. in the first equation

y=2x+4

y=2⋅1+4

y=6

The solution of the linear system is (1, 6).

You can use the substitution method even if both equations of the linear system are in standard form. Just begin by solving one of the equations for one of its variables.

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