Equações lineares com duas variáveis e seus gráficos

Summary

In this video, the concept of linear equations is explored, focusing on how they represent straight lines on a graph. The presenter demonstrates how to plot the equation y = 2x - 3 by calculating corresponding y-values for specific x-values. The relationship between x and y is shown to always produce points that lie on a straight line. Additionally, the video covers different forms of linear equations, such as 4x - 3y = 12, and contrasts them with non-linear equations like y = x². A key takeaway is the defining characteristic of a linear equation: both variables are raised to the power of one and are multiplied by constants.

Takeaways

• 😀 Equations of the line represent relationships between variables where the graph is a straight line.
• 😀 The equation y = 2x - 3 is a linear equation and describes a straight line on the Cartesian plane.
• 😀 When plotting points that satisfy the equation y = 2x - 3, a straight line is formed, as shown by the points (0, -3), (1, -1), and (2, 1).
• 😀 The line contains all the points that satisfy the equation, including fractions or decimals like (-0.5, -4).
• 😀 For a given equation, every x-value has a corresponding y-value that lies on the line.
• 😀 For example, with x = 6, the equation y = 2x - 3 gives y = 9, confirming that the point (6, 9) lies on the line.
• 😀 The equation can also be expressed in different forms, such as 4x - 3y = 12, which still represents a straight line.
• 😀 By calculating x and y values, you can plot more points on the line. For instance, when x = 0, y = -4, and when x = 3, y = 0.
• 😀 Not all equations produce straight lines. For example, y = x² or x * y = 12 form curves, not lines.
• 😀 A straight line equation follows a specific form: x and y are raised to the power of 1, and constants are multiplied by variables. No variables are raised to higher powers or divided by each other.

Q & A

What is an equation of a straight line?

An equation of a straight line is a mathematical expression where the relationship between two variables, usually x and y, can be represented as a straight line when graphed. For example, y = 2x - 3 is an equation of a straight line.

Why is the equation y = 2x - 3 called an equation of a straight line?

It is called an equation of a straight line because, when you plot the pairs of values (x, y) that satisfy the equation on a Cartesian plane, they form a straight line.

How do we determine points on the graph of the equation y = 2x - 3?

To determine points on the graph, substitute different values of x into the equation to find corresponding y values. For example, if x = 0, y = -3; if x = 1, y = -1; and if x = 2, y = 1. These points can then be plotted on the graph.

How many points are needed to draw a straight line on a graph?

To draw a straight line, you only need two points. Once you have two points, you can connect them to form a straight line. In the case of the equation y = 2x - 3, we used the points (0, -3), (1, -1), and (2, 1).

Can the straight line contain points that are not integer coordinates?

Yes, the straight line contains all points that satisfy the equation, including non-integer values. For instance, if x = -0.5, the corresponding y value is -4, which is a valid point on the line.

What happens if the x value is 6 in the equation y = 2x - 3?

When x = 6, substitute it into the equation: y = 2(6) - 3 = 12 - 3 = 9. The point (6, 9) will lie on the straight line.

Can other types of equations form straight lines?

No, not all equations form straight lines. Equations such as y = x² or x * y = 12 do not represent straight lines but rather curves.

What is the general form of a linear equation?

A linear equation typically has the form Ax + By = C, where A, B, and C are constants, and x and y are the variables raised to the power of 1. The equation does not include squared terms or products of x and y.

What is the significance of constants in a linear equation?

Constants in a linear equation, such as -3 in y = 2x - 3 or 12 in 4x - 3y = 12, represent fixed values that do not change. These constants help determine the position and slope of the line on the graph.

Why is it important that the variables in a linear equation are raised only to the power of 1?

It is important because equations where the variables are raised to the power of 1 (such as y = 2x - 3) represent straight lines. If the variables are squared or have other exponents, the graph will form curves instead of straight lines.