Problem 36 - Algebra 2

Question

Write an equation in slope-intercept form for the line that satisfies each set of conditions. passes through \((-3,-1),\) parallel to the line that passes through \((3,3)\) and \((0,6)\)

Step-by-Step Solution

Answer

The equation is \( y = -x - 4 \).

Step 1: Determine the Slope of the Original Line

To find the slope of the line passing through the points (3,3) and (0,6), use the slope formula: \( m = \frac{y_2-y_1}{x_2-x_1} \). Here, \( x_1 = 3 \), \( y_1 = 3 \), \( x_2 = 0 \), \( y_2 = 6 \). Substitute these values into the formula to get \( m = \frac{6-3}{0-3} = \frac{3}{-3} = -1 \). The slope of the original line is \(-1\).

Step 2: Use the Parallel Line Slope

Since parallel lines have the same slope, the slope of the new line is also \(-1\).

Step 3: Apply the Point-Slope Formula

Now, use the point-slope formula \( y - y_1 = m(x - x_1) \), substituting \( m = -1 \), \( x_1 = -3 \), and \( y_1 = -1 \). This gives \( y + 1 = -1(x + 3) \).

Step 4: Simplify to Slope-Intercept Form

Simplify the equation from Step 3: \( y + 1 = -1(x + 3) \). Distribute \(-1\) to get \( y + 1 = -1x - 3 \). Then, subtract 1 from both sides to obtain the slope-intercept form: \( y = -1x - 3 - 1 \). This simplifies to \( y = -x - 4 \).

Key Concepts

Slope Calculation

The concept of slope is crucial to understanding lines in mathematics. Slope measures how steep a line is. It is determined by the ratio of the change in the vertical direction (rise) to the change in the horizontal direction (run). Mathematically, the slope, often symbolized as \( m \), is calculated using the formula: \[ m = \frac{y_2-y_1}{x_2-x_1} \] Where:

• \( y_1, y_2 \) are the y-coordinates of two distinct points on the line.
• \( x_1, x_2 \) are the x-coordinates of those two points.

This formula gives us a number that describes the line's inclination. In the original problem, applying this formula to the points \((3,3)\) and \((0,6)\) allowed us to find the slope \( m = -1 \). This information is fundamental for further steps when working with linear equations.

Parallel Lines

Parallel lines are lines in a plane that never intersect. Because they run in the same direction, they have the identical slope. If you identify the slope of one line, you automatically know the slope of any parallel line. This is why the problem specifies a line parallel to the one through points \((3,3)\) and \((0,6)\): we use the same slope value. In our example, once we computed the slope of the reference line to be \(-1\), we inferred that any line parallel to it must also have a slope of \(-1\). Recognizing this symmetry in parallel lines helps simplify problems and allows the use of known slopes without recalculating them.

Point-Slope Form

The point-slope form of a line's equation is a handy way to write the equation of a line when you know a point on the line and the slope. The formula for the point-slope form is: \[ y - y_1 = m(x - x_1) \] Where:

• \( m \) is the slope of the line.
• \( (x_1, y_1) \) are the coordinates of a known point on the line.

For the problem at hand, using the point \((-3,-1)\) and slope \(-1\), we substitute these values into the point-slope formula: \[ y + 1 = -1(x + 3) \] This equation serves as an intermediary step before converting to the slope-intercept form, making it especially useful when you're gathering all components of the line before simplifying.

Equation of a Line

Determining the equation of a line and expressing it in slope-intercept form \(y = mx + b\) is a common task in algebra. This form is beneficial because it provides direct insight into the line's slope \( m \) and its y-intercept \( b \). The y-intercept is the point where the line crosses the y-axis. In this exercise, after applying the point-slope form, we simplified to slope-intercept form. Starting with: \[ y + 1 = -1(x + 3) \] Distribute the slope: \[ y + 1 = -x - 3 \] To isolate \( y \), subtract \( 1 \) from both sides: \[ y = -x - 4 \] Here, the slope, \(-1\), and y-intercept, \(-4\), are clearly revealed, providing an easy-to-interpret line equation.