Graph the line.
[tex]y=-\frac{3}{2} x+2[/tex]
Answer (2)
Identify the y-intercept: The line intersects the y-axis at ( 0 , 2 ) .
Use the slope to find another point: With a slope of − 2 3 , moving 2 units right and 3 units down from the y-intercept gives the point ( 2 , − 1 ) .
Plot the two points: Plot ( 0 , 2 ) and ( 2 , − 1 ) on the coordinate plane.
Draw the line: Draw a straight line through the two points to represent the graph of y = − 2 3 x + 2 .
y = − 2 3 x + 2
Explanation
Understanding the Problem The equation of the line is given as y = − 2 3 x + 2 . This is in slope-intercept form, y = m x + b , where m is the slope and b is the y-intercept. The slope is m = − 2 3 and the y-intercept is b = 2 . To graph the line, we need at least two points.
Finding the y-intercept The y-intercept is the point where the line crosses the y-axis, which occurs when x = 0 . In this case, the y-intercept is ( 0 , 2 ) .
Using the Slope to Find Another Point The slope of the line is − 2 3 . This means that for every 2 units we move to the right (increase in x), we move 3 units down (decrease in y). Starting from the y-intercept ( 0 , 2 ) , we can find another point on the line by moving 2 units to the right and 3 units down. This gives us the point ( 2 , − 1 ) .
Plotting the Points and Drawing the Line Now we have two points on the line: ( 0 , 2 ) and ( 2 , − 1 ) . We can plot these points on the coordinate plane and draw a straight line through them. This line represents the graph of the equation y = − 2 3 x + 2 .
Final Answer The graph of the line y = − 2 3 x + 2 passes through the points ( 0 , 2 ) and ( 2 , − 1 ) .
Examples
Understanding linear equations is crucial in many real-world applications. For instance, if you are saving money, you can model your savings with a linear equation where the slope represents your savings rate and the y-intercept represents your initial savings. Similarly, in physics, the relationship between distance, speed, and time can be modeled using a linear equation. Graphing these equations helps visualize the relationship and make predictions.
To graph the line y = − 2 3 x + 2 , identify the y-intercept at ( 0 , 2 ) and use the slope − 2 3 to find another point, such as ( 2 , − 1 ) . Plot these two points and draw a straight line through them to complete the graph.
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