Suppose x can be expressed as a function of y by x = 5y − 3.
a. How can y be expressed as a function of x?
b. What is the relationship between the slopes of the two linear equations?
c. How do the x-intercepts and y-intercepts of the two equations relate?
a)
x = 5y − 3
5y = x+3
y = (1/5)x + 3/5
b) the slope is 1/5, they are the same equation
c) they are the same equation
To express y as a function of x, we can rearrange the given equation, x = 5y - 3, to solve for y. Follow these steps:
a. Expressing y as a function of x:
1. Start with the given equation: x = 5y - 3.
2. Add 3 to both sides of the equation to isolate the term with y: x + 3 = 5y.
3. Divide both sides of the equation by 5 to solve for y: (x + 3) / 5 = y.
Therefore, y can be expressed as a function of x by the equation y = (x + 3) / 5.
b. Relationship between the slopes of the two linear equations:
The slope of a linear equation in the form y = mx + b represents the rate of change of y with respect to x. In this case, the given equation x = 5y - 3 can be rewritten in the slope-intercept form as y = (1/5)x + (3/5).
Comparing the two equations, we can see that the slope of the first equation (y = (x + 3) / 5) is 1/5, while the slope of the second equation (y = (1/5)x + (3/5)) is also 1/5. Therefore, the slopes of the two linear equations are equal.
c. Relationship between the x-intercepts and y-intercepts of the two equations:
The x-intercept of an equation is the value of x where the graph intersects the x-axis (i.e., y = 0), while the y-intercept is the value of y where the graph intersects the y-axis (i.e., x = 0).
For the first equation, y = (x + 3) / 5, to find the x-intercept, set y = 0 and solve for x:
0 = (x + 3) / 5
Multiplying both sides by 5:
0 = x + 3
Solving for x, we get x = -3. Therefore, the x-intercept is -3.
For the second equation, y = (1/5)x + (3/5), to find the y-intercept, set x = 0 and solve for y:
y = (1/5) * 0 + (3/5)
Simplifying, we get y = 3/5. Therefore, the y-intercept is 3/5.
Therefore, the x-intercept of the first equation (-3) and the y-intercept of the second equation (3/5) are related.