Understanding the Solutions to Equations: A Step-by-Step Guide
In this article, we will delve into the topic of solving two-variable linear equations. We will focus on understanding and identifying the solutions to these equations. Before we begin, let’s make sure we are all on the same page with some key definitions.
What are Two-Variable Linear Equations?
Two-variable linear equations are mathematical expressions that involve two variables, usually represented as ‘x’ and ‘y.’ These equations have the form ‘ax + by = c,’ where ‘a,’ ‘b,’ and ‘c’ are numerical coefficients.
Determining the Solutions
To find the solutions to these equations, we need to substitute different pairs of values for ‘x’ and ‘y’ and check if the equation holds true. In this article, we will analyze five pairs of values and determine which pairs satisfy the given equations.
Exploring the Examples
Let’s examine the following pairs of values: (-2; 1), (0; 2), (-1; 0), (1.5; 3), and (4; -3). We will determine which of these pairs are solutions for the equations below:
a) 5x + 4y = 8
b) 3x + 5y = -3
Analyzing Equation (a)
For equation (a), let’s substitute the different pairs of values and see if they satisfy the equation.
Pair (-2; 1):
5(-2) + 4(1) = -10 + 4 = -6 ≠ 8
Therefore, the pair (-2; 1) is not a solution of the equation 5x + 4y = 8.Pair (0; 2):
5(0) + 4(2) = 0 + 8 = 8
Therefore, the pair (0; 2) is a solution of the equation 5x + 4y = 8.Pair (-1; 0):
5(-1) + 4(0) = -5 ≠ 8
Therefore, the pair (-1; 0) is not a solution of the equation 5x + 4y = 8.Pair (1.5; 3):
5(1.5) + 4(3) = 7.5 + 12 = 19.5 ≠ 8
Therefore, the pair (1.5; 3) is not a solution of the equation 5x + 4y = 8.Pair (4; -3):
5(4) + 4(-3) = 20 – 12 = 8
Therefore, the pair (4; -3) is a solution of the equation 5x + 4y = 8.
From the analysis above, we can conclude that the pairs (0; 2) and (4; -3) are solutions of the equation 5x + 4y = 8.
Analyzing Equation (b)
For equation (b), let’s substitute the different pairs of values and see if they satisfy the equation.
Pair (-2; 1):
3(-2) + 5(1) = -6 + 5 = -1 ≠ -3
Therefore, the pair (-2; 1) is not a solution of the equation 3x + 5y = -3.Pair (0; 2):
3(0) + 5(2) = 10 ≠ -3
Therefore, the pair (0; 2) is not a solution of the equation 3x + 5y = -3.Pair (-1; 0):
3(-1) + 5(0) = -3
Therefore, the pair (-1; 0) is a solution of the equation 3x + 5y = -3.Pair (1.5; 3):
3(1.5) + 5(3) = 4.5 + 15 = 19.5 ≠ -3
Therefore, the pair (1.5; 3) is not a solution of the equation 3x + 5y = -3.Pair (4; -3):
3(4) + 5(-3) = 12 – 15 = -3
Therefore, the pair (4; -3) is a solution of the equation 3x + 5y = -3.
Hence, we can conclude that the pairs (-1; 0) and (4; -3) are solutions of the equation 3x + 5y = -3.
Wrapping Up
In this article, we have explored the concept of solving two-variable linear equations. We have analyzed examples and determined which pairs of values satisfy the given equations. Remember, the solutions to these equations can be found by substituting different pairs of values and checking if the equation holds true. By understanding and applying these principles, you can enhance your skills in solving two-variable linear equations.
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