Problem 44 - Algebra 1

Question

Write an equation in standard form of the line that passes through the given point and has the given slope. $$(-3,3), m=4$$

Step-by-Step Solution

Answer

The equation of the line in standard form that passes through the point (-3,3) and has a slope of 4 is \(4x - y = -15\).

Step 1: Use the slope-intercept form

We use the slope-intercept form of the line equation which is \(y = mx + b\), where \(m = 4\) is the slope and \((-3, 3)\) is the point that the line passes through. Substitute these values in to get the equation: \(3 = 4*(-3) + b\). From here, we solve for \(b\).

Step 2: Solve for the y-intercept

After substitution, the equation becomes \(3 = -12 + b\). Solve this to find \(b\) which gives us \(b = 15\).

Step 3: Write the equation of the line in slope-intercept form

Now that we have the slope (m = 4) and the y-intercept (b = 15), we can write the equation of the line in slope-intercept form: \(y = 4x + 15\).

Step 4: Convert into standard form

The last step is to convert the slope-intercept form to standard form. The standard form is defined as \(Ax + By = C\). Let's transform our equation \(y - 4x = 15\). However, it's common to leave standard form with \(A\) as a positive integer, so we multiply the entire equation by -1, which gives us \(4x - y = -15\).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most commonly used ways to express a line in mathematics. It's written as \(y = mx + b\). This format is very intuitive because it clearly shows two key features of the line.

• \(m\) is the slope of the line. It indicates how steep the line is. Consistent across straight lines, it's like a rate showing how much \(y\) changes for a unit change in \(x\).
• \(b\) is the y-intercept. This tells us where the line will intersect the y-axis, which means the value of \(y\) when \(x\) equals zero.

To find the slope-intercept form, identify the values of \(m\) and \(b\) and plug them into the equation.

Slope

The slope of a line is a measure of its steepness or slant. It's a ratio that compares the change in the "rise" (vertical movement) to the "run" (horizontal movement) between any two points on the line.
For the exercise, the slope \(m\) is given as 4, which can be interpreted following way:

• For every unit increase in \(x\), \(y\) will increase by 4 units.
• This slope of 4 indicates a fairly steep line.

Slope can be calculated by the formula: \( m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}\). Once the value is known, it helps in predicting the behavior of the line efficiently.

Y-Intercept

The y-intercept, represented as \(b\) in the slope-intercept form, is the point where the line crosses the y-axis. This implies that at this point, the value of \(x\) is zero. In many real-world problems, this marks the starting point or baseline when tracking changes, as it's the initial value before any variables start affecting it.
In this exercise, after substituting the given point \((-3,3)\) and the slope \(m = 4\) into the slope-intercept equation, we calculate \(b = 15\).
Hence, the line crosses the y-axis at (0, 15). This value of \(b\) often becomes crucial when transforming the equation into different forms for varied applications.

Point-Slope Formula

The point-slope formula provides a method to write the equation of a line when you know one point on the line and the slope. This is useful when a direct approach like the slope-intercept form isn't ideal because information about the y-intercept is not given. The formula is: \[ y - y_1 = m(x - x_1) \] Here, \((x_1, y_1)\) is a known point on the line, and \(m\) is the slope.
In the original exercise, we know the slope \(m = 4\) and the point \((-3, 3)\). Plug these into the point-slope form:

• \(y - 3 = 4(x + 3)\)

From this form, you can easily manipulate it to derive the equation in other forms like the slope-intercept form or even the standard form, depending on what is needed.