In algebra, slope-intercept form is a way to express linear equations in a convenient and easy-to-understand format. The standard form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. Here, we will explore five ways to express the equation 5x + 2y = 10 in slope-intercept form.
Understanding the Equation
Before we dive into the different ways to express the equation, let's understand the equation itself. The equation 5x + 2y = 10 is a linear equation in two variables, x and y. The coefficients of x and y are 5 and 2, respectively, and the constant term is 10.
Method 1: Rearranging the Terms
The first way to express the equation in slope-intercept form is to rearrange the terms. We can move the term with x to the right-hand side of the equation and the term with y to the left-hand side.
5x + 2y = 10
Subtracting 5x from both sides:
2y = -5x + 10
Dividing both sides by 2:
y = (-5/2)x + 5
This is the slope-intercept form of the equation.
Method 2: Using the Slope-Intercept Formula
Another way to express the equation in slope-intercept form is to use the slope-intercept formula. The slope-intercept formula is y = mx + b, where m is the slope and b is the y-intercept.
We can rewrite the equation as:
2y = -5x + 10
Dividing both sides by 2:
y = (-5/2)x + 5
This is the same as the previous method.
Method 3: Graphing the Equation
A third way to express the equation in slope-intercept form is to graph the equation. By graphing the equation, we can find the slope and y-intercept of the line.
The equation 5x + 2y = 10 can be graphed on a coordinate plane.
From the graph, we can see that the slope of the line is -5/2 and the y-intercept is 5.
Therefore, the equation can be expressed in slope-intercept form as:
y = (-5/2)x + 5
Method 4: Using the Point-Slope Form
A fourth way to express the equation in slope-intercept form is to use the point-slope form. The point-slope form is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
We can choose a point on the line, such as (0, 5), and use the point-slope form to find the equation.
y - 5 = (-5/2)(x - 0)
Simplifying the equation:
y = (-5/2)x + 5
This is the same as the previous methods.
Method 5: Using the Intercept Form
A fifth way to express the equation in slope-intercept form is to use the intercept form. The intercept form is x/a + y/b = 1, where a is the x-intercept and b is the y-intercept.
We can rewrite the equation as:
x/2 + y/5 = 1
Multiplying both sides by 5:
5x/2 + y = 5
Subtracting 5x/2 from both sides:
y = (-5/2)x + 5
This is the same as the previous methods.
Conclusion
In this article, we explored five ways to express the equation 5x + 2y = 10 in slope-intercept form. We used rearranging terms, the slope-intercept formula, graphing the equation, the point-slope form, and the intercept form to find the slope-intercept form of the equation. Each method yielded the same result: y = (-5/2)x + 5.
We hope this article has helped you understand the different ways to express linear equations in slope-intercept form.
What is the slope-intercept form of a linear equation?
+The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.
How do you express an equation in slope-intercept form?
+There are several ways to express an equation in slope-intercept form, including rearranging terms, using the slope-intercept formula, graphing the equation, using the point-slope form, and using the intercept form.
What is the slope of the line in the equation 5x + 2y = 10?
+The slope of the line in the equation 5x + 2y = 10 is -5/2.