solve this---**Chart Description:**
* **Type:** Cartesian coordinate system with a line graph.
* **Main Elements:**
* **Coordinate Axes:** Horizontal x-axis and vertical y-axis intersect at the origin O. Both axes have tick marks representing integer units. The positive direction of the x-axis is to the right, and the positive direction of the y-axis is upwards. The number '1' is labeled on both the x and y axes.
* **Line:** A straight line is drawn in the coordinate system. It is labeled with the letter 'l'. The line passes through the point (0, 3) on the y-axis (y-intercept) and the point (2, 0) on the x-axis (x-intercept). The line has a negative slope, extending from the upper left to the lower right.
**Question Stem:**
What is an equation of line $\ell$ in the figure above?
**Options:**
(A) $y = -\frac{3}{2}x + 2$
(B) $y = -\frac{3}{2}x + 3$
(C) $y = -\frac{2}{3}x + 3$
(D) $y = \frac{2}{3}x + 2$
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In this problem, we need to find the equation of line ℓ shown in the coordinate system. Let's start by identifying the key points on the line. Looking at the figure, we can see that the line passes through two important points: the y-intercept at (0, 3) and the x-intercept at (2, 0). These points will help us determine the equation of the line.
Now that we've identified the key points, let's calculate the slope of the line. The slope formula is m equals y2 minus y1 divided by x2 minus x1. Using our points (0, 3) and (2, 0), we get m equals 0 minus 3 divided by 2 minus 0, which equals negative 3 divided by 2, or negative three-halves. We can visualize this as a rise of negative 3 units and a run of 2 units, giving us a slope of negative three-halves.
Now that we have the slope, let's find the y-intercept to complete our equation. We'll use the slope-intercept form of a line: y equals m x plus b, where m is the slope and b is the y-intercept. We already know that m equals negative three-halves. To find b, we can substitute one of our known points. Let's use the y-intercept point (0, 3). When we substitute x equals 0 and y equals 3, we get: 3 equals negative three-halves times 0 plus b, which simplifies to 3 equals b. Therefore, b equals 3. Our final equation is y equals negative three-halves x plus 3.
Let's verify our equation y equals negative three-halves x plus 3 by checking if it passes through our known points. For the point (0, 3), we substitute x equals 0 into our equation and get y equals negative three-halves times 0 plus 3, which equals 3. This matches the y-coordinate of our point. For the point (2, 0), we substitute x equals 2 and get y equals negative three-halves times 2 plus 3, which equals negative 3 plus 3, which equals 0. This matches the y-coordinate of our second point. Now, looking at the given options, we can see that option B, y equals negative three-halves x plus 3, matches our derived equation. Therefore, the answer is B.
Let's summarize what we've learned. To find the equation of a line, we first identify two points on the line. Then we calculate the slope using the formula m equals y2 minus y1 divided by x2 minus x1. Next, we use the slope-intercept form of a line, y equals mx plus b, and substitute a known point to find the y-intercept b. For this problem, we found that the equation is y equals negative three-halves x plus 3, which corresponds to option B. It's always good practice to verify your answer by checking if the equation passes through the given points, which we did for both (0, 3) and (2, 0).