Question

The graph of a line in the \(x y\)-plane passes through the point \((-2, k)\) and crosses the \(x\)-axis at the point \((-4,0)\). The line crosses the \(y\)-axis at the point \((0,12)\). What is the value of \(k\) ? $$ 5\left(10 x^2-300\right)+\left(9,844+50 x^2\right) $$

Short answer

The value of \(k\) is 6 for the given line passing through the point \((-2, k)\), crossing the x-axis at \((-4,0)\) and the y-axis at \((0,12)\).

Step-by-step solution

Find the slope of the line

We are given two points on the line: \((-4,0)\) and \((0,12)\). To find the slope of this line, we can use the formula: $$m = \frac{y2 - y1}{x2 - x1}$$ Now, insert the values of the given points (-4,0) and (0,12) into the equation: $$m = \frac{12 - 0}{0 - (-4)}$$

Calculate the slope

Use the equation with the given values to find the slope m: $$m = \frac{12}{4}$$ $$m = 3$$ So, the slope of the line is m = 3.

Find the equation of the line

Now, we will use the point-slope form of a linear equation: $$y - y1 = m(x - x1)$$ Since the line crosses the y-axis at (0,12), we can use the point (0,12) and the slope found in Step 2: $$y - 12 = 3(x - 0)$$

Simplify the equation for the line

Simplify the equation found in Step 3 to get the line's equation in slope-intercept form (y = mx + b): $$y - 12 = 3x$$ $$y = 3x + 12$$ Now we have the equation of the line as \(y = 3x +12\).

Find the value of k

We are given the point \((-2, k)\), so we can find the value of \(k\) by substituting the x-coordinate into the line's equation: $$k = 3(-2) + 12$$ $$k = -6 + 12$$ $$k = 6$$ So, the value of \(k\) is 6, and the point is \((-2, 6)\).