Questions and Answers
What is the first step to solve a linear equation in two variables?
Graph the equation
What is the slopeintercept form of a linear equation in two variables?
y = mx + b
What is the first step to solve for x and y in the given linear equation?
Graph the equation
How can the general form Ax + By = C be rewritten to find the slope and yintercept?
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What does the slope of a linear equation in two variables represent?
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Why are linear equations important in various fields like physics and economics?
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What does the equation $y = \frac{2}{3}x + 2$ represent?
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What does solving for one variable in terms of the other variable allow us to do?
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Which principle can be used to solve for one variable in terms of the other variable in a linear equation?
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What is the next step after solving for one variable in terms of the other variable?
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Study Notes
Equation in Two Variables
An equation in two variables refers to a mathematical statement that connects two variables with the use of the equality operator (=). The variables can take any value within a specified domain, and the equation can be expressed as a straight line when plotted on a graph. In this article, we will focus on linear equations, which are equations of the first degree in two variables.
Linear Equations
A linear equation in two variables is an equation that can be written in the slopeintercept form, y = mx + b, where m is the slope and b is the yintercept. The slope represents the rate of change of the variable y with respect to the variable x, while the yintercept represents the value of y when x = 0.
The general form of a linear equation in two variables is:
$$Ax + By = C$$
Here, A, B, and C are constants, and x and y are the variables. To find the slope and yintercept of this equation, we can rewrite it in the slopeintercept form by solving for y:
$$y = \frac{A}{B}x + \frac{C}{B}$$
The slope of this equation is $$\frac{A}{B}$$, and the yintercept is $$\frac{C}{B}$$.
Solving Linear Equations
To solve a linear equation in two variables, we can use the following steps:

Graph the equation: Plot the equation on a graph and determine the domain and range of the variables.

Identify the variables: Determine which variable is dependent and which is independent. This can often be done by looking at the domain and range of the variables.

Solve for one variable: Solve for one of the variables in terms of the other variable. This can be done using the principle of substitution or elimination.

Substitute the value: Substitute the value of the variable you solved for into the equation to find the value of the other variable.
For example, consider the following linear equation:
$$2x + 3y = 6$$
To solve for x and y, we can follow these steps:

Graph the equation: Plot the equation on a graph and determine the domain and range of the variables.

Identify the variables: In this case, both x and y are dependent variables.

Solve for one variable: We can solve for y in terms of x by rearranging the equation:
$$y = \frac{2}{3}x + \frac{6}{3}$$
$$y = \frac{2}{3}x + 2$$
Now, we have solved for y in terms of x.
 Substitute the value: To find the value of y when x = 3, substitute x = 3 into the equation:
$$y = \frac{2}{3}(3) + 2$$
$$y = \frac{6}{3} + 2$$
$$y = 2 + 2$$
$$y = 0$$
So, when x = 3, y = 0.
Applications of Linear Equations
Linear equations have numerous applications in various fields, such as physics, engineering, and economics. For example, they can be used to model the relationship between two variables, such as the relationship between the cost of a product and the quantity produced. They can also be used to solve problems involving rates, such as the rate at which a car travels down a road.
In conclusion, equations in two variables, particularly linear equations, are essential tools in mathematics and various fields. They can be used to model relationships, solve problems, and make predictions. By understanding the basics of linear equations, we can apply them to a wide range of realworld situations and make informed decisions.
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Description
Learn about linear equations in two variables and how to solve them using the slopeintercept form. Explore the applications of linear equations in various fields like physics, engineering, and economics. Understand the steps involved in solving linear equations and how they can be graphed to determine the domain and range of the variables.