Find the slope of the line whose equation is $8 y=2 x+4$.
Answer (1)
Rewrite the given equation 8 y = 2 x + 4 in slope-intercept form.
Divide both sides of the equation by 8 to isolate y : y = 8 2 x + 8 4 .
Simplify the equation to y = 4 1 x + 2 1 .
Identify the slope as the coefficient of x , which is 4 1 .
Explanation
Understanding the Problem We are given the equation of a line: 8 y = 2 x + 4 . Our goal is to find the slope of this line. To do this, we need to rewrite the equation in slope-intercept form, which is y = m x + b , where m represents the slope and b represents the y-intercept.
Isolating y To convert the given equation to slope-intercept form, we need to isolate y on one side of the equation. We can do this by dividing both sides of the equation by 8: 8 8 y = 8 2 x + 4
Simplifying the Equation Simplifying the equation, we get: y = 8 2 x + 8 4 Now, we can further simplify the fractions: y = 4 1 x + 2 1
Identifying the Slope Now that the equation is in slope-intercept form ( y = m x + b ), we can identify the slope, m . In this case, m = 4 1 . Therefore, the slope of the line is 4 1 .
Final Answer The slope of the line 8 y = 2 x + 4 is 4 1 .
Examples
Understanding the slope of a line is crucial in many real-world applications. For example, in construction, the slope of a ramp determines its steepness and accessibility. In economics, the slope of a supply or demand curve indicates how sensitive the quantity supplied or demanded is to changes in price. By finding the slope, we can analyze and predict relationships between variables in various fields.
