The Table Represents The Equation $y = 2 - 4x$.$\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline -2 & 10 \\ \hline -1 & \\ \hline 0 & 2 \\ \hline 1 & -2 \\ \hline 2 & -6 \\ \hline \end{tabular} \\]Use The Drop-down Menus To
The Equation of a Line: Understanding the Relationship Between x and y
In mathematics, the equation of a line is a fundamental concept that describes the relationship between two variables, x and y. The equation of a line is typically represented in the form y = mx + b, where m is the slope of the line and b is the y-intercept. In this article, we will explore the equation of a line, specifically the equation y = 2 - 4x, and use a table to represent the relationship between x and y.
The Equation y = 2 - 4x
The equation y = 2 - 4x is a linear equation that describes a straight line. The slope of the line is -4, which means that for every unit increase in x, y decreases by 4 units. The y-intercept of the line is 2, which means that when x is equal to 0, y is equal to 2.
Using a Table to Represent the Equation
A table can be used to represent the equation y = 2 - 4x by listing the values of x and the corresponding values of y. The table below represents the equation y = 2 - 4x.
| x | y |
|---|---|
| -2 | 10 |
| -1 | |
| 0 | 2 |
| 1 | -2 |
| 2 | -6 |
Finding the Missing Value
The table above has a missing value for x = -1. To find the missing value, we can substitute x = -1 into the equation y = 2 - 4x.
y = 2 - 4(-1) y = 2 + 4 y = 6
Therefore, the missing value for x = -1 is y = 6.
The Completed Table
The completed table is shown below.
| x | y |
|---|---|
| -2 | 10 |
| -1 | 6 |
| 0 | 2 |
| 1 | -2 |
| 2 | -6 |
Understanding the Relationship Between x and y
The table above shows the relationship between x and y for the equation y = 2 - 4x. As x increases, y decreases. This is because the slope of the line is negative, which means that y decreases as x increases.
Graphing the Equation
The equation y = 2 - 4x can be graphed on a coordinate plane. The graph of the equation is a straight line with a slope of -4 and a y-intercept of 2.
In conclusion, the equation y = 2 - 4x is a linear equation that describes a straight line. The table above represents the relationship between x and y for the equation. The missing value for x = -1 was found by substituting x = -1 into the equation. The completed table shows the relationship between x and y for the equation. The graph of the equation is a straight line with a slope of -4 and a y-intercept of 2.
The equation y = 2 - 4x has many applications in real-life situations. For example, the equation can be used to model the relationship between the cost of a product and the number of units sold. The equation can also be used to model the relationship between the temperature of a substance and the amount of heat added to it.
Real-World Examples
- Cost of a Product: Suppose a company sells a product for $10 per unit. The cost of producing the product is $2 per unit, and the company wants to make a profit of $4 per unit. The equation y = 2 - 4x can be used to model the relationship between the cost of the product and the number of units sold.
- Temperature of a Substance: Suppose a substance is heated from 0°C to 100°C. The equation y = 2 - 4x can be used to model the relationship between the temperature of the substance and the amount of heat added to it.
Linear equations can be solved using various methods, including substitution and elimination. The equation y = 2 - 4x can be solved using substitution by substituting y = 2 - 4x into the equation.
Substitution Method
- Substitute y = 2 - 4x into the equation: y = 2 - 4x
- Solve for x: x = (y - 2) / 4
Elimination Method
- Multiply both sides of the equation by 4: 4y = 8 - 16x
- Add 16x to both sides of the equation: 4y + 16x = 8
- Divide both sides of the equation by 4: y + 4x = 2
Q: What is the equation y = 2 - 4x?
A: The equation y = 2 - 4x is a linear equation that describes a straight line. The slope of the line is -4, which means that for every unit increase in x, y decreases by 4 units. The y-intercept of the line is 2, which means that when x is equal to 0, y is equal to 2.
Q: How do I use the equation y = 2 - 4x to find the value of y when x is equal to a certain value?
A: To find the value of y when x is equal to a certain value, substitute the value of x into the equation y = 2 - 4x. For example, if x is equal to 1, substitute x = 1 into the equation to get y = 2 - 4(1) = -2.
Q: What is the slope of the line described by the equation y = 2 - 4x?
A: The slope of the line described by the equation y = 2 - 4x is -4. This means that for every unit increase in x, y decreases by 4 units.
Q: What is the y-intercept of the line described by the equation y = 2 - 4x?
A: The y-intercept of the line described by the equation y = 2 - 4x is 2. This means that when x is equal to 0, y is equal to 2.
Q: How do I graph the equation y = 2 - 4x on a coordinate plane?
A: To graph the equation y = 2 - 4x on a coordinate plane, plot the y-intercept (0, 2) and then use the slope (-4) to find other points on the line. For example, if x is equal to 1, y is equal to -2, so plot the point (1, -2) on the coordinate plane.
Q: What are some real-world applications of the equation y = 2 - 4x?
A: The equation y = 2 - 4x has many real-world applications, including modeling the relationship between the cost of a product and the number of units sold, and modeling the relationship between the temperature of a substance and the amount of heat added to it.
Q: How do I solve the equation y = 2 - 4x using the substitution method?
A: To solve the equation y = 2 - 4x using the substitution method, substitute y = 2 - 4x into the equation and then solve for x. For example, if y is equal to 1, substitute y = 1 into the equation to get 1 = 2 - 4x, and then solve for x.
Q: How do I solve the equation y = 2 - 4x using the elimination method?
A: To solve the equation y = 2 - 4x using the elimination method, multiply both sides of the equation by 4 to get 4y = 8 - 16x, and then add 16x to both sides of the equation to get 4y + 16x = 8. Then, divide both sides of the equation by 4 to get y + 4x = 2.
Q: What is the relationship between the slope and the y-intercept of the line described by the equation y = 2 - 4x?
A: The slope and the y-intercept of the line described by the equation y = 2 - 4x are related in that the slope is the negative reciprocal of the y-intercept. In this case, the slope is -4 and the y-intercept is 2, so the negative reciprocal of 2 is -1/2, but the slope is -4, which is the negative reciprocal of 1/4, but the equation is y = 2 - 4x, so the slope is -4 and the y-intercept is 2.
Q: How do I use the equation y = 2 - 4x to model the relationship between the cost of a product and the number of units sold?
A: To use the equation y = 2 - 4x to model the relationship between the cost of a product and the number of units sold, substitute the cost of the product into the equation for y and the number of units sold into the equation for x. For example, if the cost of the product is $10 and the number of units sold is 2, substitute y = 10 and x = 2 into the equation to get 10 = 2 - 4(2), and then solve for the number of units sold.
Q: How do I use the equation y = 2 - 4x to model the relationship between the temperature of a substance and the amount of heat added to it?
A: To use the equation y = 2 - 4x to model the relationship between the temperature of a substance and the amount of heat added to it, substitute the temperature of the substance into the equation for y and the amount of heat added into the equation for x. For example, if the temperature of the substance is 100°C and the amount of heat added is 4 units, substitute y = 100 and x = 4 into the equation to get 100 = 2 - 4(4), and then solve for the amount of heat added.