Problem 8 - Calculus

Question

Determine the slope and the \(y\) -intercept of the line whose equation is given. $$2 y+5 x-8=0$$

Step-by-Step Solution

Answer

Slope is \(-\frac{5}{2}\) and y-intercept is 4.

Step 1: Rearrange the Equation

The given equation is \(2y + 5x - 8 = 0\). To find the slope and y-intercept, we need to write it in the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Start by isolating \(2y\) by moving the other terms to the right side of the equation: \(2y = -5x + 8\).

Step 2: Solve for y

Divide every term in the equation \(2y = -5x + 8\) by 2 to solve for \(y\):\[ y = -\frac{5}{2}x + 4 \] This is now in the form \(y = mx + b\).

Step 3: Identify the Slope

From the equation \(y = -\frac{5}{2}x + 4\), compare it with the slope-intercept form \(y = mx + b\). The coefficient of \(x\) (\(-\frac{5}{2}\)) is the slope \(m\).

Step 4: Identify the y-Intercept

From the rearranged equation \(y = -\frac{5}{2}x + 4\), the constant term (\(4\)) is the y-intercept \(b\).

Key Concepts

Linear Equation

A linear equation is what we often see in math. It's simply an equation that describes a straight line on a graph. Linear equations typically look like this: \( ax + by + c = 0 \), where \( a \), \( b \), and \( c \) are constants. In the exercise, the linear equation is given as \(2y + 5x - 8 = 0\).

• Linear means that if you were to graph the equation, it would appear as a straight line.
• The equation includes variables (like \(x\) and \(y\)) and constants (like \(2\), \(5\), and \(-8\)).

Linear equations help us understand the relationship between two variables, often showing how a change in one variable impacts another. When presented in standard form, we can rearrange it into the slope-intercept form for easier understanding.

Slope

In the world of mathematics, the slope is a crucial part of any linear equation. It describes how steep a line is on a graph. You might also think of it as the angle or tilt of the line. In the equation \(y = mx + b\), \(m\) represents the slope.

• A positive slope means the line goes upwards as it moves from left to right.
• A negative slope means the line goes downwards as it moves from left to right.

For our exercise, we found the slope to be \(-\frac{5}{2}\). This tells us the line slants downward. The \( \frac{5}{2} \) indicates that for every 2 units you move horizontally to the right on the graph, you will move 5 units downwards.The slope helps us quickly understand the direction and steepness of the line.

Y-Intercept

The y-intercept is a vital component of a linear equation. It's the point where the line crosses the y-axis on a graph. In \(y = mx + b\), \(b\) is the y-intercept.

• It is the value of \(y\) when \(x\) is 0.
• This point can tell us a lot about the line itself, such as its starting point on the vertical axis.

In our example, \(y\)-intercept is 4, meaning the line crosses the \(y\)-axis at \( (0, 4) \). This point serves as an anchor from where we can measure the slope direction. Whenever you're trying to visualize or understand a linear equation, starting with the y-intercept makes it easier.