Solving Simultaneous Equations Graphically
Solving simultaneous equations graphically means drawing both equations as lines. The crossing point solves both equations. There is one solution if lines cross, no solution if parallel, and infinite solutions if they overlap.
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Simultaneous Linear Equations
- When and form a set of simultaneous equations, they are solved together.
- We look for x and y values that make both equations true at the same time.
Solving Simultaneous Equations Graphically
- Rearrange both equations into y = mx + c form.
- Plot both lines and find where they intersect.
- The point of intersection is the solution of the simultaneous equations.
Solutions of Simultaneous Linear Equations
- There is one solution if the lines intersect once.
- There is no solution if the lines are parallel.
- There are infinite solutions if the lines overlap exactly.
Practice Questions
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How many solutions do the following simultaneous equations have?
Correct! 🎉 +10 pointsNot quite right
When we graph both equations, we see that the lines are parallel (because they have the same gradient of 2) and do not intersect. This means there is no point where both equations are satisfied simultaneously. In other words, there is no solution to these simultaneous equations.
How many solutions do the following simultaneous equations have?
Correct! 🎉 +10 pointsNot quite right
When we graph both equations, we see that the lines are parallel (because they have the same gradient of -1) and do not intersect. This means there is no point where both equations are satisfied simultaneously. In other words, there is no solution to these simultaneous equations.
How many solutions do the following simultaneous equations have?
Correct! 🎉 +20 pointsNot quite right
When we graph both equations, we see that the lines intersect at one point, meaning there is a unique solution. In this case, the point of intersection is (1, 3), so the simultaneous equations have one solution.
How many solutions do the following simultaneous equations have?
Correct! 🎉 +20 pointsNot quite right
When we graph both equations, we see that the lines intersect at one point, meaning there is a unique solution. In this case, the point of intersection is (4, 2), so the simultaneous equations have one solution.
How many solutions do the following simultaneous equations have?
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If we double the first equation, we get exactly the second equation. When we graph the two equations, the lines overlap perfectly. Every point on the line satisfies both equations. Therefore, the simultaneous equations have infinitely many solutions.
Emily and James’ combined age is 30 years. Emily is 6 years older than James. What are their ages?
Correct! 🎉 +30 pointsNot quite right
First, let's form the simultaneous equations: , where x is Emily's age and y is James' age, and , since Emily is 6 years older than James. Next, solving the simultaneous equations by graphing, we find the point of intersection at and . Therefore, Emily is 18 years old, and James is 12 years old.
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Interactive Activity
Find the intersection of the two lines
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Students Also Ask
The questions students bump into most on this topic
Solving them graphically means drawing each equation as a straight line on the same coordinate grid. The point where the lines cross gives the values that satisfy both equations. You read off the x and y coordinates of that intersection, and those coordinates are the solution to the pair.
Every point on a line satisfies that line's equation. The intersection is the one point that lies on both lines at once, so it satisfies both equations together. That is exactly what a solution to simultaneous equations means, which is why the crossing point gives you the answer.
Yes. Rearrange each equation into the form y = mx + c first, making y the subject. This lets you read the gradient and the y-intercept straight from the equation, which makes each line quick to plot. Once both lines are drawn, you can find where they cross.
If the two lines are parallel, they never cross, so the simultaneous equations have no solution. This happens when both equations have the same gradient but different y-intercepts. Because the lines stay the same distance apart forever, no single point can satisfy both equations at once.
They have infinitely many solutions when the two lines lie exactly on top of each other. This happens when the equations are identical after you rearrange them. Every point on the shared line satisfies both equations, so every one of those points is a valid solution to the pair.