Problem 32 - Algebra and Trigonometry

Question

Find an equation of the line that satisfies the given conditions. y-intercept 6 : parallel to the line \(2 x+3 y+4=0\)

Step-by-Step Solution

Answer

Equation of the line: \( y = -\frac{2}{3}x + 6 \).

Step 1: Find the slope of the given line

Rewrite \(2x + 3y + 4 = 0\) in slope-intercept form: \(y = -\frac{2}{3}x - \frac{4}{3}\). The slope is \(m = -\frac{2}{3}\).

Step 2: Use the same slope for the parallel line

Parallel lines have the same slope. With \(m = -\frac{2}{3}\) and \(y\)-intercept \(b = 6\):
\(y = -\frac{2}{3}x + 6\)

Key Concepts

Slope-Intercept Form

The slope-intercept form is a common way to write the equation of a line. It looks like this: \[ y = mx + b \]- **\(y\)** represents the y-coordinate of any point on the line. - **\(m\)** stands for the slope, which is the line's steepness. - **\(x\)** is the x-coordinate of any point on the line. - **\(b\)** is the y-intercept, where the line crosses the y-axis. This form makes it easy to identify the slope and y-intercept directly from the equation. For example, in the equation \( y = -\frac{2}{3}x - \frac{4}{3} \), you can immediately see that the slope \(m\) is \(-\frac{2}{3}\) and the y-intercept \(b\) is \(-\frac{4}{3}\). Using this form can simplify many math problems, such as finding parallel or perpendicular lines, graphing linear equations, and solving algebraic equations involving lines. It's particularly useful in real-world problems where the rate of change (slope) and initial value (y-intercept) are important.

Parallel Lines

Parallel lines are lines in a plane that never meet. They always have the same slope. This makes them very easy to spot when written in slope-intercept form. Consider two lines: - **Line 1:** \( y = mx + b_1 \) - **Line 2:** \( y = mx + b_2 \) Both lines have the same slope \(m\), making them parallel. The only difference is in the y-intercept \(b\), which determines where each line crosses the y-axis. In our exercise, when asked to find a line parallel to \(2x + 3y + 4 = 0\), we first identified its slope to be \(-\frac{2}{3}\). Since parallel lines share the same slope, the new line must also have a slope of \(-\frac{2}{3}\). We will set its y-intercept as given to complete the equation of the desired line. Parallel lines are frequently used in geometry and algebra problems due to their unique properties. Recognizing parallel lines is crucial in solving systems of equations and designing geometric shapes.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis. It's represented by \(b\) in the slope-intercept form \( y = mx + b \). To find the y-intercept in an equation, you can simply look at the \(b\) value when the equation is in slope-intercept form. This is the value of \(y\) when \(x = 0\). In the given problem, the y-intercept of the line parallel to \(2x + 3y + 4 = 0\) is specified as 6. This tells us that our new line will cross the y-axis at the point \((0, 6)\). Knowing the y-intercept is a critical step in graphing lines and understanding their positions in relation to other lines. It helps in visualizing how high or low the line starts on the graph. This initial value often represents starting conditions in many real-world scenarios, such as initial costs or starting positions.