Mada za sehemu hiiUse basic coordinate geometry, trigonometry, and vectors skills in daily lifeMada 5
- Explore the basic tenets of coordinate geometry (gradient and equations of a straight line, graphs of linear equations)
- Find the gradient/slope of a line
- Determine the equation of a straight line and draw its graph
- Solve linear simultaneous equations graphically
- Use mathematical software to solve and draw graphs of simultaneous equations
Solving Linear Simultaneous Equations Graphically
When you have two equations with two unknown numbers (usually x and y), and both equations must be true at the same time, you have simultaneous equations. The graphical method solves them by drawing both lines on a coordinate plane and finding where they cross — that crossing point is the solution.
Step 1: Rewrite each equation in the form y = mx + c (or solve for y).
Step 2: Find where each line crosses the x-axis and y-axis (the intercepts).
Step 3: Plot the two intercepts on the graph and draw a straight line through them. Do the same for the second equation.
Step 4: Read the coordinates of the point where the two lines intersect. This point (x, y) is the solution.
Step 5: Check your answer by substituting the values into both original equations.
Solve the following simultaneous equations graphically:
Solution
For the first line: x + y = 5
- When x = 0, y = 5 → point (0, 5)
- When y = 0, x = 5 → point (5, 0)
For the second line: x - y = 1
- When x = 0, -y = 1, so y = -1 → point (0, -1)
- When y = 0, x = 1 → point (1, 0)
Plot these points and draw both lines. They intersect at the point (3, 2).
Check:
- First equation: 3 + 2 = 5 ✓
- Second equation: 3 - 2 = 1 ✓
Therefore, the solution is x = 3 and y = 2.
- If two lines are parallel (never meet), there is no solution.
- If two lines lie on top of each other (same line), there are infinitely many solutions.
- Always verify your graphical answer using substitution to ensure accuracy.
In Tanzania, a farmer might use simultaneous equations to decide how many bags of maize and beans to sell at the market. For example, if a trader offers two different price combinations that total a certain amount of Tanzanian shillings, plotting these relationships on a graph helps the farmer find exactly how many bags of each to sell to meet the target income. This graphical method makes it easy to visualize different possibilities and choose the best option.
Swali
What does the point of intersection of two lines represent when solving simultaneous equations graphically?
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