Problem 40 - College Algebra with Modeling and Visualization

Question

Find the slope-intercept form for the line satisfying the conditions. Parallel to \(y=-4 x-\frac{1}{4},\) passing through \((2,-5)\)

Step-by-Step Solution

Answer

The slope-intercept form is \(y = -4x + 3\).

Step 1: Analyze the Given Equation

The given line equation is in the slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope. The equation \(y = -4x - \frac{1}{4}\) has a slope of \(-4\).

Step 2: Use the Slope of the Parallel Line

For two lines to be parallel, they must have the same slope. Therefore, the slope of the line we are determining is also \(-4\).

Step 3: Use the Point-Slope Equation

Utilize the point-slope form \(y - y_1 = m(x - x_1)\) to write the equation of the line. Here, \((x_1, y_1) = (2, -5)\) is a point on our line. Substitute \(m = -4\) into this formula: \(y + 5 = -4(x - 2)\).

Step 4: Simplify to Get the Slope-Intercept Form

Distribute the slope on the right side and simplify: \(y + 5 = -4(x - 2)\) \(y + 5 = -4x + 8\). Subtract \(5\) from both sides to solve for \(y\): \(y = -4x + 3\).

Step 5: Confirm the Slope-Intercept Form

Check the final equation \(y = -4x + 3\) to ensure it has retained the parallel slope \(-4\) and passes through the point \((2, -5)\). Substitute \(x = 2\) to get \(y = -4(2) + 3 = -8 + 3 = -5\), which is correct.

Key Concepts

parallel lines

Parallel lines are lines in a plane that never meet. No matter how far you extend them, they stay at a consistent distance apart. This consistency means they have the same slope. The slope is a measure of steepness and is the same for both lines. For example, if a line has an equation of the form

• \(y = -4x - \frac{1}{4}\),

then any line parallel to it must also have a slope of

• \(-4\).

When trying to find a parallel line, simply use the same slope in your equation.

point-slope equation

The point-slope equation is extremely useful for finding 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, \((x_1, y_1)\) represents a specific point on the line, and \(m\) is the slope. For example, if you know a line passes through the point \((2, -5)\) and has a slope of -4, plug these values into the point-slope formula to obtain:

• \(y - (-5) = -4(x - 2)\),

which simplifies to

• \(y + 5 = -4(x - 2)\).

algebra

In algebra, we manipulate equations to solve for one variable. When given an equation, such as one in point-slope form, our goal is often to convert it into slope-intercept form \(y = mx + b\). This form makes it easy to read off the slope \(m\) and the y-intercept \(b\). To convert

• \(y + 5 = -4(x - 2)\)

to slope-intercept form, we perform distributive property and simplify:

• \(y + 5 = -4x + 8\).

Subtract 5 from both sides:

• \(y = -4x + 3\).

Now the equation is in a simplified form showing the slope is -4 and the y-intercept is 3.

coordinate geometry

Coordinate geometry allows us to represent geometric shapes using an algebraic approach. Lines can be depicted on a two-dimensional plane using equations. Each line is defined by its slope and a point it passes through. When determining where lines intersect or are parallel, we use the equations of these lines.For example, a line parallel to a given line on the plane has the same slope as the original. Using coordinate geometry, we can visually and algebraically establish that two lines never touch by showing they always have the same slope.By knowing a point

• \((2, -5)\),

through which the new line passes, alongside the slope of -4, coordinate geometry lets us clearly derive the line's equation and depict how it fits into the overall plane structure.