Find The Slope Of The Line In Three Ways For The Equation $5x + 2y = 10$.(a) Give Any Two Points That Lie On The Line And Use Them To Determine The Slope. (b) Solve The Equation For $y$, And Identify The Slope From The Equation. (c)

Introduction

The slope of a line is a fundamental concept in mathematics, representing the rate of change of a function or the steepness of a line. In this article, we will explore three methods to find the slope of a line given by the equation $5x + 2y = 10$. We will use these methods to determine the slope of the line and provide a comprehensive understanding of each approach.

Method (a): Using Two Points

To find the slope of a line using two points, we need to identify any two points that lie on the line. Let's solve the equation $5x + 2y = 10$ for $y$ to find the $y$-intercept, which will give us one point on the line.

Step 1: Solve the equation for y

To solve the equation for $y$, we need to isolate $y$ on one side of the equation. We can do this by subtracting $5x$ from both sides and then dividing both sides by $2$.

$$5x + 2y = 10$$

$$2y = -5x + 10$$

$$y = -\frac{5}{2}x + 5$$

Now that we have the equation in slope-intercept form, we can identify the $y$-intercept, which is $(0, 5)$. This is one point on the line.

Step 2: Find another point on the line

To find another point on the line, we can choose any value of $x$ and substitute it into the equation to find the corresponding value of $y$. Let's choose $x = 2$.

$$y = -\frac{5}{2}(2) + 5$$

$$y = -5 + 5$$

$$y = 0$$

So, another point on the line is $(2, 0)$.

Step 3: Calculate the slope using the two points

Now that we have two points on the line, we can calculate the slope using the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

where $(x_1, y_1)$ and $(x_2, y_2)$ are the two points on the line.

$$m = \frac{0 - 5}{2 - 0}$$

$$m = \frac{-5}{2}$$

So, the slope of the line is $-\frac{5}{2}$.

Method (b): Solving the Equation for y

In this method, we will solve the equation $5x + 2y = 10$ for $y$ to find the slope of the line.

Step 1: Solve the equation for y

To solve the equation for $y$, we need to isolate $y$ on one side of the equation. We can do this by subtracting $5x$ from both sides and then dividing both sides by $2$.

$$5x + 2y = 10$$

$$2y = -5x + 10$$

$$y = -\frac{5}{2}x + 5$$

Now that we have the equation in slope-intercept form, we can identify the slope, which is $-\frac{5}{2}$.

Method (c): Using the Slope-Intercept Form

In this method, we will use the slope-intercept form of a line to find the slope.

Step 1: Identify the slope-intercept form

The slope-intercept form of a line is given by:

$$y = mx + b$$

where $m$ is the slope and $b$ is the $y$-intercept.

Step 2: Identify the slope

In the equation $y = -\frac{5}{2}x + 5$, the slope is $-\frac{5}{2}$.

Conclusion

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Q: What is the slope of a line?

A: The slope of a line is a measure of how steep the line is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

Q: How do I find the slope of a line using two points?

A: To find the slope of a line using two points, you need to identify the coordinates of the two points and then use the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

where $(x_1, y_1)$ and $(x_2, y_2)$ are the two points on the line.

Q: How do I find the slope of a line by solving the equation for y?

A: To find the slope of a line by solving the equation for y, you need to isolate y on one side of the equation. The slope will be the coefficient of x in the equation.

Q: What is the slope-intercept form of a line?

A: The slope-intercept form of a line is given by:

$$y = mx + b$$

where m is the slope and b is the y-intercept.

Q: How do I find the slope of a line using the slope-intercept form?

A: To find the slope of a line using the slope-intercept form, you need to identify the coefficient of x in the equation. This will be the slope of the line.

Q: What is the difference between the slope and the y-intercept?

A: The slope is a measure of how steep the line is, while the y-intercept is the point where the line intersects the y-axis.

Q: Can I find the slope of a line if I only know the y-intercept?

A: No, you cannot find the slope of a line if you only know the y-intercept. You need to know at least two points on the line or the equation of the line in slope-intercept form to find the slope.

Q: Can I find the slope of a line if I only know the x-intercept?

A: No, you cannot find the slope of a line if you only know the x-intercept. You need to know at least two points on the line or the equation of the line in slope-intercept form to find the slope.

Q: How do I graph a line if I know the slope and the y-intercept?

A: To graph a line if you know the slope and the y-intercept, you can use the slope-intercept form of the line:

$$y = mx + b$$

where m is the slope and b is the y-intercept. You can plot the y-intercept on the graph and then use the slope to determine the direction and steepness of the line.

Q: How do I graph a line if I know two points on the line?

A: To graph a line if you know two points on the line, you can plot the two points on the graph and then draw a line through them. The slope of the line can be calculated using the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

where $(x_1, y_1)$ and $(x_2, y_2)$ are the two points on the line.