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Brilio.net - Linear equations of two variables are one of the basic concepts in mathematics a that is often used in various applications, both in everyday life and in academic fields. In its most general form, this equation can be written as ax+by=c, where a, b, and c are constants, and x and y are the variables whose values you want to find. The main characteristic of this equation is that the exponent of each variable is one, which makes it linear.
The use of linear equations in two variables is very broad. In the context of economics, for example, these equations are often used to calculate profits or production costs. In the context of education, understanding these equations is essential for students to prepare for exams and understand further concepts in algebra. Therefore, it is important to master how to solve and apply linear equations in two variables.
There are several ways to solve linear equations in two variables, including substitution, elimination, and graphical methods. The substitution method involves replacing one variable with the value of another variable, while the elimination method focuses on eliminating one variable to find the value of the other. By understanding these methods, students can more easily solve problems related to linear equations in two variables.
Explanation of Linear Equations of Two Variables
Example of a linear equation with two variables
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A linear equation in two variables is a mathematical system consisting of two equations with two variables. Its general form is:
ax + by = c
Where:
- a and b are the coefficients of the variables x and y.
- c is a constant.
- The variables x and y have a power of one.
This equation has many applications in everyday life, such as in calculating the price of goods, budgeting, and data analysis. To solve this system of equations, we can use several methods such as substitution and elimination.
Example of Linear Equations of Two VariablesExample question 1
Given the following system of equations:
1. 2x + 3y = 12
2. x y = 1
Question:
Find the value of
x and y.
Completion
Step 1: substitution or elimination
We can use the substitution or elimination method. Here, we will use the substitution method.
From the second equation, we can express x in terms of y:
x = y + 1
Step 2: substitute into the first equation
Now, we substitute the value of x into the first equation:
2 (y+1) + 3y = 12
Step 3: Simplify the Equation
Let's simplify the equation:
2y + 2 + 3y = 12
Combine like terms:
5y + 2 = 12
Step 4: Isolate Variables
5y = 10
Then, divide both sides by 5:
y = 2
Step 5: Find the Value of x
Now we substitute the value of y back into the equation to find x:
x = y + 1 = 2 + 1 = 3
The final result
So, the solution to the system of equations is:
x = 3,y = 2
Conclusion
The x and y values that satisfy both equations are x = 3 and y = 2.
Example question 2
Given the following system of equations:
1. 3x + 4y = 24
2. 2x y = 3
Question:
Find the values of x and y.
Completion
Step 1: isolate variables
From the second equation, we can express y in terms of x:
y = 2x 3
Step 2: substitute into the first equation
Now, substitute the value of y into the first equation:
3x + 4 (2x3) = 24
Step 3: simplify the equation
Let's simplify it:
3x + 8x 12 = 24
Combine like terms:
11 x 12 = 24
Step 4: Isolate the Variable x
Add 12 to both sides:
11 x = 36
Then, divide both sides by 11:
x= 11/36
Step 5: Find the value of y
Substitute the value of x back into the equation to find y:
y = 2 ( 36/11) 3 = 72/11 33/11 = 39/11
The final result
So, the solution to the system of equations is:
x = 36/11, y = 39/11
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