solve this---**Image Description:**
The image contains a 2D Cartesian coordinate system with the x-axis and y-axis intersecting at the origin O. The positive x-axis points to the right, and the positive y-axis points upwards. A straight line, labeled l, is drawn on the coordinate plane, extending from the upper left down into the lower right.
There are three labeled points or reference points shown. A point is labeled on the y-axis with coordinates (0, 4). A point is labeled on the x-axis with coordinates (3, 0). A third point is marked on the line l. Dashed lines extend horizontally from the point (0, 4) to the point on line l, and vertically from the point on line l down to the x-axis at the x-coordinate corresponding to the point (3, 0). Right angle symbols are shown at the points where the dashed lines meet the axes, indicating perpendicularity. These dashed lines and labels indicate that the line l passes through the point with x-coordinate 3 and y-coordinate 4, i.e., the point (3, 4).
An annotation below the figure states, "Note: Figure not drawn to scale."
**Question:**
In the figure above, if line l has a slope of -2, what is the y-intercept of l ?
**Options:**
(A) 7
(B) 8
(C) 9
(D) 10
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In this problem, we need to find the y-intercept of line l. We're given that the line has a slope of negative 2 and passes through the point (3, 4). Let's set up the coordinate system and visualize the problem. The line l passes through the point (3, 4) as shown in the figure. We need to find where this line intersects the y-axis.
To find the y-intercept, we'll use the slope-intercept form of a line, which is y equals m x plus b, where m is the slope and b is the y-intercept we're looking for. We know that the slope m equals negative 2, and the line passes through the point (3, 4). Let's substitute these values into the equation. We get 4 equals negative 2 times 3 plus b. The slope triangle shows how a slope of negative 2 means that for every 1 unit we move right, we move down 2 units.
Now let's solve for the y-intercept b. We have the equation 4 equals negative 2 times 3 plus b. First, let's simplify the right side: 4 equals negative 6 plus b. To isolate b, we add 6 to both sides of the equation: 4 plus 6 equals b, which gives us b equals 10. So the y-intercept of line l is 10. We can verify this by writing the equation of the line in slope-intercept form: y equals negative 2x plus 10. Looking at our options, the correct answer is D, 10.
Let's summarize how we found the y-intercept of line l. We used the slope-intercept form of a line, which is y equals m x plus b, where m is the slope and b is the y-intercept. Given that the slope m equals negative 2 and the line passes through the point (3, 4), we substituted these values into the equation to get 4 equals negative 2 times 3 plus b. Simplifying, we got 4 equals negative 6 plus b, which gave us b equals 10. Therefore, the y-intercept of line l is 10, and the correct answer is option D.
Let's explore an alternative method to find the y-intercept using the point-slope form of a line. The point-slope form is y minus y₁ equals m times x minus x₁, where m is the slope and (x₁, y₁) is a point on the line. Substituting our known values, slope m equals negative 2 and the point (3, 4), we get y minus 4 equals negative 2 times x minus 3. Expanding the right side, we get y minus 4 equals negative 2x plus 6. Adding 4 to both sides, we get y equals negative 2x plus 10. So the y-intercept is still 10, confirming our answer is option D.