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Systems of Linear Equations

Systems of Linear Equations. Substitution. Linear Equations. There are 3 different ways to solve linear equations: 1. Substitution 2. Elimination 3. Graphing We will focus on a new one each day. Today is Substitution. Substitution. Substitution is just what it sounds like.

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Systems of Linear Equations

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  1. Systems of Linear Equations Substitution

  2. Linear Equations • There are 3 different ways to solve linear equations: • 1. Substitution • 2. Elimination • 3. Graphing • We will focus on a new one each day. • Today is Substitution.

  3. Substitution • Substitution is just what it sounds like. • Rewrite one equation and substitute it into the other equation. • x + y = 270 • 6x + 8y = 2080 • y = 270 – x • 6x + 8(270 – x) = 2080 • 6x + 2160 – 8x = 2080 • 2160 – 2x = 2080 • - 2x = - 80 • x = 40 • y = 270 – 40 • y = 230 • (40, 230) Solve one equation for a single variable. Substitute the “new” equation into the 2ns equation. Solve for the variable. Plug that answer back into one of the equations and solve for the 2nd variable.

  4. Substitution • When solving systems of equations, solutions will fall into one of three categories. • 1. One Solution • 2. No solution • 3. Infinitely many solutions • One solution means that the lines cross one time. That is the intersection we are solving for. • No solution means the lines never cross….parallel lines. • Infinitely many solutions means the equations are for the same lines.

  5. Substitution • Solve: • x – y = 2 • 2x – 2y = 10 • x = 2 + y • 2(2 + y) – 2y = 10 • 4 + 2y – 2y = 10 • 4 = 10 ??? • This is no solution because 4 does not equal 10.

  6. Substitution • Solve: • x – y = 2 • - x + y = - 2 • x = 2 + y • - (2 + y ) + y = - 2 • - 2 – y + y = - 2 • - 2 = - 2 • This is true, so infinitely many solutions.

  7. Substitution • Your turn to try a few! • 2x + y = 3 x – y = 2 x + 2y = 1 • 4x + 2y = 4 4x – 3y = 10 2x + 4y = 2 • No Solution (4, 2) Infinitely many solutions

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