- Isolate \(y\) in the first equation, \(y - 5 = -4x\), by adding 5 to both sides to get \(y = -4x + 5\).
- Isolate \(y\) in the second equation, \(3y - 9 = -6x\), by adding 9 to both sides to get \(3y = -6x + 9\), then divide by 3 to get \(y = -2x + 3\).
- The slope-intercept form of the first equation is \(\boxed{y = -4x + 5}\).
- The slope-intercept form of the second equation is \(\boxed{y = -2x + 3}\).
Explanation
- Analysis of the Linear Equations
We are given a system of two linear equations in the variables \(x\) and \(y\):
1) \(y - 5 = -4x\)
2) \(3y - 9 = -6x\)
Our objective is to rewrite both equations in the slope-intercept form, which is given by the formula \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept. This form is particularly useful because it explicitly shows the rate of change and the starting point (the y-intercept) of each linear relationship.
- Converting the First Equation to Slope-Intercept Form
To convert the first equation \(y - 5 = -4x\) into slope-intercept form, we need to isolate the variable \(y\) on the left side. We do this by adding 5 to both sides of the equation:
\[y - 5 + 5 = -4x + 5\]
\[y = -4x + 5\]
This equation is now in the form \(y = mx + b\), where the slope \(m = -4\) and the y-intercept \(b = 5\).
- Converting the Second Equation to Slope-Intercept Form
To convert the second equation \(3y - 9 = -6x\) into slope-intercept form, we again isolate the variable \(y\). First, we add 9 to both sides:
\[3y - 9 + 9 = -6x + 9\]
\[3y = -6x + 9\]
Next, to isolate \(y\) completely, we divide every term in the equation by 3:
\[\frac{3y}{3} = \frac{-6x}{3} + \frac{9}{3}\]
\[y = -2x + 3\]
This equation is now in the form \(y = mx + b\), where the slope \(m = -2\) and the y-intercept \(b = 3\).
- Final Results
By performing algebraic manipulation on both given equations, we have successfully expressed them in the requested slope-intercept form.
The first equation is \(y = -4x + 5\).
The second equation is \(y = -2x + 3\).
The final answers are:
The slope-intercept form of the first equation is \(\boxed{y = -4x + 5}\).
The slope-intercept form of the second equation is \(\boxed{y = -2x + 3}\).
Examples
The slope-intercept form is widely used in physics and economics to represent linear relationships between two variables. For example, if you are tracking the speed of a vehicle, the equation \(y = mx + b\) can represent the total distance \(y\) covered in time \(x\), where \(m\) is the constant speed (slope) and \(b\) is the initial distance (y-intercept) from a specific point. Understanding how to rearrange equations into this form allows you to easily interpret and graph real-world data to make predictions or analyze trends.