Consider the system of equations.
[tex]y=3 x+2[/tex]
[tex]y=-\frac{2}{3} x-4[/tex]
Explain why these particular equations can be graphed immediately.
These can be graphed immediately because the [tex]x[/tex]-axis is not unknown and we know where to plot both the points.
Answer (2)
The equations can be graphed immediately because they are in slope-intercept form, which gives the slope and y-intercept directly. From this form, we can easily plot points and draw the lines on a graph. This makes understanding their relationship simple and efficient.
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The equations are in slope-intercept form, y = m x + b .
Identify the slope m and y-intercept b from each equation.
Use the y-intercept to plot a point on the line.
Use the slope to find additional points and draw the line.
Explanation
Understanding the Problem We are given a system of two linear equations:
y = 3 x + 2
y = − 3 2 x − 4
The question asks us to explain why these equations can be graphed immediately. The provided reason is that the x-axis is not unknown and we know where to plot both the points. Let's analyze this.
Recognizing Slope-Intercept Form The key to understanding why these equations can be graphed immediately lies in recognizing their form. Both equations are presented in slope-intercept form, which is:
y = m x + b
where:
m represents the slope of the line
b represents the y-intercept (the point where the line crosses the y-axis)
Identifying Slopes and Y-Intercepts In our system of equations:
For the first equation, y = 3 x + 2 , the slope m = 3 and the y-intercept b = 2 . This means the line crosses the y-axis at the point (0, 2).
For the second equation, y = − 3 2 x − 4 , the slope m = − 3 2 and the y-intercept b = − 4 . This means the line crosses the y-axis at the point (0, -4).
Graphing with Slope and Y-Intercept Because we can directly read off the slope and y-intercept from the equations, we can easily graph these lines. The y-intercept gives us one point on the line, and the slope tells us how to find other points. For example, a slope of 3 means that for every 1 unit we move to the right on the x-axis, we move 3 units up on the y-axis.
The provided reason that the x-axis is not unknown and we know where to plot both the points is partially correct, but not complete. The x-axis is indeed known, but the main reason we can immediately graph these equations is because they are in slope-intercept form, which directly provides the slope and y-intercept.
Final Answer In conclusion, the equations y = 3 x + 2 and y = − 3 2 x − 4 can be graphed immediately because they are in slope-intercept form ( y = m x + b ). This form allows us to easily identify the slope ( m ) and y-intercept ( b ) of each line, which are sufficient to plot the lines on a coordinate plane.
Examples
Imagine you're designing a ramp. The equation of the ramp's slope can be expressed in slope-intercept form. Knowing the slope and y-intercept allows you to quickly visualize and construct the ramp. Similarly, in financial planning, linear equations can represent investment growth. The slope indicates the rate of return, and the y-intercept represents the initial investment. Understanding these parameters helps in making informed decisions.
