Grade 10 → Algebra → Linear Equations in Two Variables ↓
Solutions of Linear Equations
When we talk about linear equations in math, we're looking at equations that represent straight lines. Linear equations in two variables are equations with two different variables, usually represented by x and y. These equations are in the form:
ax + by + c = 0
where a, b and c are real numbers, and at least one of a or b is not zero.
Understanding solutions of linear equations
A "solution" of a linear equation in two variables is a pair of values (x, y) - that satisfies the equation. We can say that a value of x paired with a value of y is a solution if it makes both sides of the equation equal when substituted into the equation. Let's consider a simple equation as an example:
2x + 3y = 6
For this particular equation, any pair of numbers (x, y) that satisfies this equation is a solution. To understand this better, let's find the solutions and plot them on a graph. This way, it will be clear how linear equations work.
Finding a solution by substitution
One way to find solutions to an equation is by substitution. Substitute different values of x and then solve for y:
Example
Let's start by substituting x = 0 into the equation:
2(0) + 3y = 6
0 + 3y = 6
3y = 6
y = 2
Hence one solution is (0, 2).
Now, substitute x = 3:
2(3) + 3y = 6
6 + 3y = 6
3y = 0
y = 0
The second solution is (3, 0).
Finally, try x = -3:
2(-3) + 3y = 6
-6 + 3y = 6
3y = 12
y = 4
The second solution is (-3, 4).
Graphical representation
When you plot the solutions of a linear equation on a two-dimensional graph (xy graph), these solutions must lie on a straight line that represents the equation itself. The solutions we just found can be plotted as points on the graph:
In this graph:
- A vertical line is the y-axis, and a horizontal line is the x-axis.
- The straight line is the graphical representation of the equation
2x + 3y = 6. - The red dots are the solutions we found earlier:
(0,2)(3,0)(-3,4)
Infinite solutions on the line
It is important to understand that linear equations have infinite solutions, not just the ones we obtain manually. For a linear equation in two variables, every point on the line (except where it extends) is a valid solution. This infinite solution set occurs because a line on the plane goes on indefinitely in both directions.
Key concepts of solution of linear equations
1. Slope-intercept form
The slope-intercept form of a linear equation is expressed as:
y = mx + b
Here, m is the slope of the line, which shows how slanted the line is, and b is the y-intercept, which is where the line crosses the y-axis. Let's rewrite our equation in slope-intercept form:
2x + 3y = 6
Solution for y:
3y = -2x + 6
y = -(2/3)x + 2
In this form, you can easily read the slope (m = -2/3) and the y-intercept (b = 2) of the line.
2. Intercept form
The intercept form of the equation involves finding the points where the line intersects the x-axis and the y-axis:
x/a + y/b = 1
We already know how to derive this: let's express our equation in intercept form:
2x + 3y = 6
To find the x-intercept where y=0:
2x = 6
x = 3
To find the y-intercept where x=0:
3y = 6
y = 2
Hence the intercept form is:
x/3 + y/2 = 1
3. Standard form
The standard form of the linear equation is given as:
ax + by = c
Our example, 2x + 3y = 6, is already in standard form, where A=2, B=3, and C=6. This form is useful for solving systems of equations but does not highlight the intercepts or slope.
Practical example
Let's work on some practical examples to deepen our understanding.
Example 1
Find solutions for:
4x – 2y = 8
Step 1: Choose a value for x, say, x=0.
4(0) – 2y = 8
-2y = 8
y = -4
One solution is (0, -4).
Step 2: Choose the other value, x=2.
4(2) – 2y = 8
8 – 2y = 8
-2y = 0
y = 0
The second solution is (2, 0).
Step 3: Graph these points to confirm linearity.
Example 2
Imagine a scenario where you are tasked with preparing a budget plan. You have to balance the expenses of rent and utilities. Let's say your total monthly budget is:
Rent + Utilities = $1200
It can be represented by the linear equation:
R + U = 1200
Choose a value for R (rent) and find U (utilities):
If the rent is $800:
800 + U = 1200
U = 400
Its solution is (800, 400).
If the rent is $600:
600 + U = 1200
U = 600
Its solution is (600, 600).
It emphasizes that real-world problems can also be formulated and solved with the help of linear equations.
Conclusion
The concept of solutions of linear equations in two variables is not only fundamental in algebra but is also widely applicable in various fields such as engineering, physics, economics, and everyday problem-solving scenarios. Understanding the nuances of finding solutions graphically and algebraically enriches understanding and strengthens the ability to tackle complex real-world challenges.
Final thoughts
By engaging with many examples, drawing diagrams, and conceptualizing problems in a linear framework, one gains confidence and clarity in using linear equations as a tool for analysis and decision making.
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