Graph the line.
y=-\frac{2}{3} x+3
Answer (2)
The equation is in slope-intercept form. The y-intercept is 3, so plot the point (0, 3). The slope is -2/3, so from the y-intercept, move 3 units to the right and 2 units down to find the point (3, 1). Draw a line through these two points. The final answer is the graph of the line y = − 3 2 x + 3 .
Explanation
Understanding the Problem The problem asks us to graph the line given by the equation y = − 3 2 x + 3 . This is a linear equation in slope-intercept form, which is y = m x + b , where m represents the slope and b represents the y-intercept.
Identifying Slope and Y-intercept In our equation, y = − 3 2 x + 3 , we can identify the slope m = − 3 2 and the y-intercept b = 3 . The y-intercept is the point where the line crosses the y-axis, which occurs when x = 0 . So, the y-intercept is the point ( 0 , 3 ) .
Finding Another Point Using the Slope To graph the line, we need at least two points. We already have one point, the y-intercept ( 0 , 3 ) . We can use the slope to find another point. The slope is − 3 2 , which means for every 3 units we move to the right (increase in x), we move 2 units down (decrease in y). Starting from the y-intercept ( 0 , 3 ) , if we move 3 units to the right, we get x = 3 . The corresponding y-value will be y = 3 − 2 = 1 . So, another point on the line is ( 3 , 1 ) .
Plotting the Points and Drawing the Line Now we have two points: ( 0 , 3 ) and ( 3 , 1 ) . We can plot these two points on a coordinate plane and draw a straight line through them. This line represents the equation y = − 3 2 x + 3 .
Final Answer The line passes through the points (0, 3) and (3, 1). The slope is negative, so the line goes downwards as we move from left to right.
Examples
Understanding linear equations and their graphs is crucial in many real-world applications. For instance, if you are tracking the depreciation of a car's value over time, you might use a linear equation to model the decrease in value each year. The slope would represent the rate of depreciation, and the y-intercept would represent the initial value of the car. Graphing this line allows you to visualize the car's value at any point in time.
To graph the line y = − 3 2 x + 3 , plot the y-intercept at (0, 3) and use the slope of -\frac{2}{3} to find another point, which is (3, 1). Draw a line through these points to complete the graph. The line should slope downward as you move from left to right.
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