Students are expected to apply multiple strategies in solving systems of equations.
While students are expected to solve most systems of equations algebraically, it is important that they be able to connect the solution with the graphical representation of the system.
Technology [T] has been identified as one of the mathematical processes to be addressed in completing this outcome. For some systems of equations, technology may be a more efficient way to solve the systems.
Strategies selected should be appropriate for the system of equations being solved.
The systems of equations may include the equations of horizontal or vertical lines.
In Achievement Indicator 9.1, students are expected to select variables that are contextually appropriate.
In Achievement Indicator 9.7, students are expected to explain why they chose a particular strategy to solve a system of linear equations.
3.9.1 Model a situation, using a system of linear equations.
Model a given situation.
Solve a given complex situation in which additional information may be needed.
3.9.2 Relate a system of linear equations to the context of a problem.
Give a partial explanation.
Give a complete explanation, including any restrictions on the domain and range.
3.9.3 Determine and verify the solution of a system of linear equations graphically, with and without technology.
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3.9.4 Explain the meaning of the point of intersection of a system of linear equations.
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3.9.5 Determine and verify the solution of linear equations algebraically.
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3.9.6 Explain, using examples, why a system of equations may have no solution, on solution, or an infinite number of solutions.
Give a partial explanation.
Give a complete explanation.
3.9.7 Explain a strategy to solve a system of linear equations.
Explain a strategy.
Explain a strategy and justify the choice of method.
3.9.8 Solve a problem that involves a system of linear equations.
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9.1 Solving Systems of Linear Equations by Substitution
Focus on...
Solving systems of linear equations algebraically using substitution.
Substitution method
An algebraic method of solving a system of equations
Solve one equation for one variable, substitute that value into the other equation, and solve for the other variable.
You can solve systems of linear equations algebraically using substitution.
Isolate a single variable in one of the two equations.
Where possible, choose a variable with a coefficient of 1. (1x or x or 1y or y)
Substitute the solution of the first variable into on of the original equations. Solve for the remaining variable.
Check your answer by substituting into both original equations.
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9.2 Solving Systems of Linear Equations by Elimination
Focus on...
Write equivalent equations to eliminate a variable.
Solving systems of linear equations algebraically using elimination
Elimination method
An algebraic method of solving a system of equations.
Add or subtract the equations to eliminate one variable and solve for the other variable.
A table can help you organize information in a problem. This can help you to determine the equations in a linear system.
You can solve a linear system by elimination
If necessary, rearrange the equations so that like variables appear in the same position in both equations. The most common form is ax + by = c.
Determine which variable to eliminate. If necessary, multiply one or both equations by a constant to eliminate the variable by addition or subtraction.
Solve for the second variable by substituting the value for the first variable into one of the original equations.
Check your solution by substituting each value into both original equations.
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9.3 Solving Problems Using Systems of Linear Equations
Focus on...
Choosing a strategy to solve a problem that involves a system of linear equations.
Substitution - look for x= or y=, or coefficients of 1
Elimination - look for the same coefficient in each equation
Graphing - no coefficients of one or same coefficient in each. Must be rearranged to y=
Solve by
Elimination when you have two coefficients the same. (2x, 2x)
Substitution when you have a coefficient of one. (Y or X)
Graphing if the equations have y isolate (y=... )
Systems of linear equations can be solved
Graphically
Algebraically by substitution or by elimination
It may be better to use a graphical approach to solve linear equations when you wish to see how the two variables relate, such as for cost analysis and speed problems.
It may be better to use an algebraic approach to solve linear equations when
You need only the solution (intersection point).
It is unclear where to locate the solution on a coordinate plane.