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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8.1 Systems of Linear Equations and Graphs
Focus on...
Explaining the meaning of the point of intersection of two linear equations
Solving systems of linear equations by creating graphs, with and without technology
Verifying solutions to systems of linear equations using substitution
Point of intersection
A point at which two lines touch or cross
System of linear equations
Two or more linear equations involving common variables
Solution (to a system of linear equations)
A point of intersection of the lines on a graph
An ordered pair that satisfies both equations
A pair of values occurring in the tables of values of both equations
Systems of linear equations can be modelled numerically, graphically or algebraically.
The solution to a linear system is a pair of values that occurs in each table of values, an intersection point of the lines on a graph, or an ordered pair that satisfies each equation.
One way to solve a system of linear equations is to graph the lines and identify the point of intersection on the graph.
A solution to a system of linear equations can be verified using several methods:
Substitute the value of each variable and evaluate the equations.
Create a graph and identify the point of intersection.
Create tables of value and identify the pair of values that occurs in each table.
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8.2 Modelling and Solving Linear Systems
Focus on...
Translating word problems into systems of linear equations
Interpreting information from the graph of a linear system
Solving problems involving systems of linear equations
When modelling word problems, assign variables that are meaningful to the context of the problem.
To assist in visualizing or organizing a word problem, you can use a diagram and /or a table of values.
If a situation involves quantities that change at constant rates, you can represent it using a system of linear equations.
If you know the initial values and rates, you can write the equations directly in slope-intercept form because the intial value is the y-intercept and the rate is the slope. Otherwise, you can determine the rate of change using start and end values.
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8.3 Number of Solutions for Systems of Linear Equations
Focus on...
Explaining why systems of linear equations can have different numbers of solutions.
Identifying how many solutions a system of linear equations has
Solving problems involving linear systems with different numbers of solutions.
When solving a system of linear equations, there are three possible options for the results:
Intersect at one point (most common)
Intersect at an infinite number of points (coincident lines)
No intersection (no solution, parallel lines)
Coincident lines
Lines that occupy the same position
In a graph of two coincident lines, any point of either line lies on the other line.
A system of linear equations can have one solution, no solution or an infinite number of solutions.
Before solving, you can predict the number of solutions for a linear system by comparing the slopes and y-intercepts.
For some linear systems, reducing the equations to lowest terms and comparing the coefficients of the x-terms, y-terms, and constants may help you predict the number of solutions.