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Solving a System Solving a System of Equationsof EquationsLinear and Linear InequalitiesLinear and Linear Inequalities

A set of two or more equations in two or A set of two or more equations in two or more variablesmore variables

Linear system- variables in each Linear system- variables in each equation are all to the power of oneequation are all to the power of one

Inequality- the equal sign has been Inequality- the equal sign has been replaced with less than, less than or replaced with less than, less than or equal to, greater than, or greater than or equal to, greater than, or greater than or equal toequal to

What is a system?What is a system?

A set of values that satisfy all the A set of values that satisfy all the equations in the system.equations in the system.

Solution SetSolution Set

3 Methods3 Methods Solving by substitution- solve one Solving by substitution- solve one

equation for a variable and plug into the equation for a variable and plug into the otherother

Solving by elimination- adding or Solving by elimination- adding or subtracting one equation from the othersubtracting one equation from the other

Graphing- graphing both equations and Graphing- graphing both equations and looking for intersection pointslooking for intersection points

Methods of SolvingMethods of Solving

Three possibilities for the number of Three possibilities for the number of solutions in a two equation system with solutions in a two equation system with two different variablestwo different variables No solution No solution One solutionOne solution Infinitely many solutionsInfinitely many solutions

Linear SystemsLinear Systems

1. Solve one equation for x or y.1. Solve one equation for x or y. 2. Substitute the expression for x or y 2. Substitute the expression for x or y

into the other equationinto the other equation 3. Solve for the remaining variable3. Solve for the remaining variable 4. Substitute the value found in Step 3 4. Substitute the value found in Step 3

into one of the original equations, and into one of the original equations, and solve for the other variablesolve for the other variable

5. Verify the solution in each equation5. Verify the solution in each equation

Solving by SubstitutionSolving by Substitution

1. Multiply one or both of the equations 1. Multiply one or both of the equations by a nonzero constant so that the by a nonzero constant so that the coefficients of x or y are opposites of one coefficients of x or y are opposites of one anotheranother

2. Eliminate x or y by adding the 2. Eliminate x or y by adding the equations, and solve for the remaining equations, and solve for the remaining variablevariable

3. Substitute the value found in step 2 3. Substitute the value found in step 2 into one of the original equations and into one of the original equations and solve for the other variablesolve for the other variable

4. Verify the solution in each equation4. Verify the solution in each equation

Solving by EliminationSolving by Elimination

{{One Step FurtherOne Step Further

Word ProblemsWord Problems

Read the problemRead the problem Define variablesDefine variables Write out the two equations firstWrite out the two equations first Solve using substitution, graphing, or Solve using substitution, graphing, or

eliminationelimination

Linear System Word Linear System Word ProblemsProblems

A ball game is attended by 575 people and total ticket sales A ball game is attended by 575 people and total ticket sales are $2575. If tickets cost $5 for adults and $3 for children, are $2575. If tickets cost $5 for adults and $3 for children, how many adults and how many children attended the gamehow many adults and how many children attended the game

A café sells two kinds of coffee in bulk. The Costa Rican sells for A café sells two kinds of coffee in bulk. The Costa Rican sells for $4.50 per pound and the Kenyan sells for $7.00 per pound. The $4.50 per pound and the Kenyan sells for $7.00 per pound. The owner wishes to mix a blend that would sell for $5.00 per pound. owner wishes to mix a blend that would sell for $5.00 per pound. How much of each type of coffee should be used in the blend?How much of each type of coffee should be used in the blend?

A toy company makes dolls, as well as collector cases for A toy company makes dolls, as well as collector cases for each doll. To make x cases costs the company $5000 in each doll. To make x cases costs the company $5000 in fixed overhead, plus $7.50 per case. An outside supplier fixed overhead, plus $7.50 per case. An outside supplier has offered to produce any desired volume of cases for has offered to produce any desired volume of cases for $8.20 per case. $8.20 per case. Write an equation that expresses the company’s cost to make x Write an equation that expresses the company’s cost to make x

casescases Write an equation that expresses the cost of buying x cases Write an equation that expresses the cost of buying x cases

from the outside supplierfrom the outside supplier When should the company make cases themselves, and when When should the company make cases themselves, and when

should they buy them from the outside supplier?should they buy them from the outside supplier?