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 \((-2, 6)\).

Key concepts

Slope Calculation

The slope of a line in coordinate geometry represents how steep the line is. It is a measure of the vertical change per unit of horizontal change as you move along the line. The formula to calculate the slope (denoted as \(m\)) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

In this exercise, the points given are

• \((-4, 0)\) — where the line crosses the \(x\)-axis.
• \((0, 12)\) — where the line crosses the \(y\)-axis.

Plugging in these points into the formula gives us:

• \(m = \frac{12 - 0}{0 - (-4)}\)
• This simplifies to \(m = \frac{12}{4} = 3\).

This calculation shows that for every unit increase in \(x\), the value of \(y\) increases by 3 units, resulting in a positive slope.

Point-Slope Form

The point-slope form is a way to describe the equation of a line when you know the slope and a point on the line. The formula is:

• \(y - y_1 = m(x - x_1)\)

Here, \((x_1, y_1)\) is a point on the line, and \(m\) is the slope. In this scenario, with the line crossing the \(y\)-axis at \((0, 12)\) and a slope of \(3\), we apply these to the point-slope form:

• \(y - 12 = 3(x - 0)\)

The expression \(x - 0\) simplifies to just \(x\), giving a ready-to-simplify equation. This form is particularly useful as it quickly gives us a starting equation from known values.

Equation of a Line

To find the full equation of a line in the slope-intercept form, which is \(y = mx + b\), we simplify the equation from the point-slope form.

• Starting from: \(y - 12 = 3x\)
• Add 12 to both sides to isolate \(y\): \(y = 3x + 12\).

This gives us the slope-intercept form, where \(m = 3\) (the slope) and \(b = 12\) (the \(y\)-intercept). The equation \(y = 3x + 12\) shows how \(y\) changes with \(x\) and provides a linear relationship between these variables, making it easy to graph.

Intercepts in Coordinate Geometry

Intercepts are crucial for understanding where a line crosses the axes.

The \(x\)-intercept is where the line crosses the \(x\)-axis. This occurs when \(y = 0\). In this exercise, it's given as \((-4, 0)\).

The \(y\)-intercept occurs where the line crosses the \(y\)-axis, at the point where \(x = 0\). Here, it crosses at \((0, 12)\).

• To find the \(x\)-intercept from an equation like \(y = 3x + 12\), set \(y = 0\):
• \(0 = 3x + 12\)
• Solve for \(x\): \(x = -4\)

This confirms the intercept at \((-4, 0)\) is correct. Intercepts provide easy reference points for graphing lines and help visualize the line's direction and position in the coordinate plane.