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Systems of linear equations in three variables
This document discusses solving systems of linear equations in three variables. It provides an example of a system of three equations with three unknowns (x, y, z). It shows the steps to solve the system using the linear combination method by eliminating a variable in two equations, solving the resulting system, then substituting back to find the values of the remaining variables. It also provides an example word problem involving three unknown amounts of money and sets up a system of equations to solve for how much each person has.
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Systems of linear equations in three variables
- 1. Systems of LinearEquations in Three Variables
- 2. Objective: Students solvesystems of linear equations and inequalities in three variables by substitution, with graphs.
- 3. A linear equationin three variables x, y, and z is an equation of the form ax + by + cz = d, where a, b, and c are not all zero. The following is an example of a system of three linear equations in three variables: 2x + y – z = 5 3x – 2y + z = 16 4x + 3y – 5z = 3
- 4. Solve Systems ofLinear Equations in Three Variables
- 5. Example 1: Solvethe system using the Linear Combination Method.
- 6. Step 1: Eliminateone of the variables in two of the original equations.
- 7. Step 1: Eliminateone of the variables in two of the original equations.
- 8. Step 2: Solvethe new system of equations
- 9. Then, Substitute z= 4 into new equation 1 or 2 and solve for x
- 10. After that, step3: Substitute x = -3 and z = 4 into any of the original equations and solve for y.
- 13. Since 0 ≠8 the result is a false equation. This system has NO SOLUTIONS.
- 14.
- 15. Solve the system: X + y + 2z = 9 2x+ 4y -3z =1 3x+ 6y -5z =0 Then find the value of x + y + z
- 16. solving a wordproblem with 3 unknowns using a linear equation Example. Amanda, Henry, and Scott have a total of $89 in thier wallets. Amanda has 6$ less than Scott. Henry has 3 times what scott has. How much does each have?
- 17. Answer Let x bethe amount of money Amanda has Let y be the amount of money Henry has Let z be the amount of money Scott has Amanda, Henry, and Scott have a total of $89 in their wallets The above statement gives the following equation x + y + z = 89 Amanda has 6$ less than Scott The above statement gives the following equation x = z - 6 Henry has 3 times what scott has. The above statement gives the following equation y = 3z
- 18. Continue... Replace x =z - 6 and y = 3z in equation 1 z - 6 + 3z + z = 89 5z - 6 = 89 5z - 6 + 6 = 89 + 6 5z = 95 z = 19 y = 3z = 3 × 19 = 57 13 = x So, Scott has 19 dollars, Henry has 57 dollars, and Amanda has 13 dollars.