Solving Systems of Linear Equations using Substitution

Substitution Any system of linear equations in two variables can be solved by the substitution method. This method lends itself to situations in which one equation is already solved for either x or y.

Substitution x = 3y + 2 2x - 4y = 10 Notice that the first equation gives us an alternate way to express x (3y + 2). Let’s re-write the second equation with the “new name” for x. 2( 3y + 2 ) – 4y = 10

Solve for the First Variable 2( 3y + 2 ) – 4y = 10 6y + 4 – 4y = 10 2y + 4 = 10 2y = 10 – 4 2y = 6 y = 3 Distribute

Solve for the Second Variable From our first equation, we know: x = 3y + 2 We just found that y = 3, so we can substitute this information into the equation to solve for x. x = 3( 3 ) + 2 x = 9 + 2 x = 11

Answer is an Ordered Pair We have found: x = 11 and y = 3 Our answer is the point of intersection of the two lines: ( 11, 3 )

Remember the BIG Picture When we are solving systems of simultaneous linear equations, we are actually looking for the point of intersection of two lines. Although a logical way to do this is by graphing, sometimes the numbers do not lend themselves very well to the graphing technique. It is almost impossible to read the point of intersection from a graph when fractions are involved. Substitution will always work.

3 Cases Just remember that there are always three possibilities when you are looking for the point of intersection of two lines. The lines can intersect in a single point. The lines can be parallel and not intersect at all. The lines can live one on top of the other with an infinite number of points of intersection.

Intersecting Lines If a system has one, or more solutions, it is said to be consistent. If the equations represent two different lines, the equations are said to be independent. If the lines intersect: the system of equations is consistent. the equations are independent. there is exactly one solution – an ordered pair the solution will be the point of intersection (x, y)

Intersecting Lines If the lines intersect in a single point, when you use the substitution method to solve the equations you will get a number for x and a number for y. Since these numbers represent the point of intersection of the two lines, they should be written as an ordered pair. In our example, the answer should be written (11, 3). Some authors, however, simply write x = 11 and y = 3.

Coincident Lines When you are solving a system of equations using the algebraic method of substitution, if you get an identity (statement that’s always true), then the lines are coincident (one on top of the other). y = 2x 5y = 10x 5(2x) = 10 x 10x = 10x

Coincident Lines If a system has one, or more solutions, it is said to be consistent. If the equations represent the same line, the equations are said to be dependent. If the lines are the same (coincident): the system of equations is consistent. the equations are dependent. there are infinite solutions – all the points on the lines sometimes the solution will be expressed in set notation. {(x, y)|y = 2x}

The two equations represent parallel lines. When you are solving a system of equations using the algebraic method of substitution, if you get an contradiction (statement that’s never true), then the lines are parallel. y = 2x 6x – 3y = - 3 6x – 3( 2x ) = - 3 6x – 6x = - 3 0 = - 3 The two equations represent parallel lines.

Parallel Lines If a system has no solutions it is said to be inconsistent. If the equations represent two different lines, the equations are said to be independent. If the lines are parallel: the system of equations is inconsistent. the equations are independent. There is no solution. In set notation, it is the empty set.

Systems of Linear Equations If you substitute the information from one equation into another, and you can solve for one variable, then use that value to solve for the other variable. That’s your point of intersection. If you get an identity, the lines are coincident. If you get a contradiction, the lines are parallel.