11A_Problem Solving with Graphs Simultaneous Equations 
Practical Applications:
Graphing Simultaneous Equations
−
−
−
−
Relating linear graphs and simultaneous equations
Analysing graphs
Practical applications of linear graphs
Writing algebraic equations
Jane Stratton
Objectives:
• Use linear graphs to solve simultaneous equations
• Use graphs of linear equations to solve a range of problems • Translate worded problems into graphical and algebraic form Finding the Solution to an Equation from a graph
• Finding solutions to an equation when we have a graph is easy, we just need to find the coordinates of points on the line. ? = ?? − ?
• Example:
? = 5 is the solution of 2? − 5 = 5
? = 4 is the solution of 2? − 5 = 3
? = 2 is the solution of 2? − 5 = −1
? = 1 is the solution of 2? − 5 = −3
? = 0 is the solution of 2? − 5 = −5
Simultaneous Equations and Graphs
• Remember: Simultaneous equations are solved at the same time – they are two equations with the same solutions. • Solving simultaneous equations using a graph is easy when you remember that the solution is where the ? and ? values are the same for both lines!
• This means you need to draw the lines of both the equations on the same graph.
• The point where the lines cross (intersect) is the solution! Example:
Solve these simultaneous equations using the graphical method:
?? + ?? = ?
?? − ?? = −?
Pick two easy numbers to plot for each equation (they’re linear so 2 points is enough!) i.e. ? = 0 and ? = 0
(0,2) and (3,0)
2
(0, ) and (-1,0)
3
Plot points and join, find the coordinates where the two lines intersect (cross).
4
(1, )
3
So, the solutions are:
4
? = ? and ? =
3
4
(1, 3)
Special Cases:
• When the number of equations is the same as the number of variables there is likely to be a solution, but this is not guaranteed.
• There are actually three possible cases:
(Infinitely many)
Applications:
• Some worded problems may require us to construct algebraic equations in order to plot a graph that can be used to solve the equations.
• Other questions
Graphing Simultaneous Equations
−
−
−
−
Relating linear graphs and simultaneous equations
Analysing graphs
Practical applications of linear graphs
Writing algebraic equations
Jane Stratton
Objectives:
• Use linear graphs to solve simultaneous equations
• Use graphs of linear equations to solve a range of problems • Translate worded problems into graphical and algebraic form Finding the Solution to an Equation from a graph
• Finding solutions to an equation when we have a graph is easy, we just need to find the coordinates of points on the line. ? = ?? − ?
• Example:
? = 5 is the solution of 2? − 5 = 5
? = 4 is the solution of 2? − 5 = 3
? = 2 is the solution of 2? − 5 = −1
? = 1 is the solution of 2? − 5 = −3
? = 0 is the solution of 2? − 5 = −5
Simultaneous Equations and Graphs
• Remember: Simultaneous equations are solved at the same time – they are two equations with the same solutions. • Solving simultaneous equations using a graph is easy when you remember that the solution is where the ? and ? values are the same for both lines!
• This means you need to draw the lines of both the equations on the same graph.
• The point where the lines cross (intersect) is the solution! Example:
Solve these simultaneous equations using the graphical method:
?? + ?? = ?
?? − ?? = −?
Pick two easy numbers to plot for each equation (they’re linear so 2 points is enough!) i.e. ? = 0 and ? = 0
(0,2) and (3,0)
2
(0, ) and (-1,0)
3
Plot points and join, find the coordinates where the two lines intersect (cross).
4
(1, )
3
So, the solutions are:
4
? = ? and ? =
3
4
(1, 3)
Special Cases:
• When the number of equations is the same as the number of variables there is likely to be a solution, but this is not guaranteed.
• There are actually three possible cases:
(Infinitely many)
Applications:
• Some worded problems may require us to construct algebraic equations in order to plot a graph that can be used to solve the equations.
• Other questions