Problem 13 - Basic Engineering Mathematics

Question

The equation of a straight line, of gradient \(m\) and intercept on the \(y\)-axis \(c\), is \(y=m x+c\). If a straight line passes through the point where \(x=1\) and \(y=-2\), and also through the point where \(x=3 \frac{1}{2}\) and \(y=10 \frac{1}{2}\), find the values of the gradient and the \(y\)-axis intercept.

Step-by-Step Solution

Answer

The gradient is 5 and the y-intercept is -7.

Step 1: Identify the Given Points and Formula

We have two points: \((1, -2)\) and \((3.5, 10.5)\). We need to find the gradient \(m\) and the intercept \(c\) for the equation of the line \(y=mx+c\).

Step 2: Calculate the Gradient

The gradient \(m\) is calculated using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute \((x_1, y_1) = (1, -2)\) and \((x_2, y_2) = (3.5, 10.5)\) into the formula: \[m = \frac{10.5 - (-2)}{3.5 - 1} = \frac{10.5 + 2}{2.5} = \frac{12.5}{2.5} = 5\].

Step 3: Use one Point to Solve for the Intercept

We use the point \((1, -2)\) and the gradient \(m = 5\) to solve for \(c\). Substitute these values into the line equation \(y = mx + c\): \(-2 = 5(1) + c\). So \(-2 = 5 + c\).

Step 4: Solve for the y-Intercept

Rearrange the equation \(-2 = 5 + c\) to find \(c\): \(c = -2 - 5 = -7\).

Step 5: State the Final Equation of the Line

With \(m = 5\) and \(c = -7\), the equation of the line is \(y = 5x - 7\).

Key Concepts

Understanding Gradient

In the realm of linear equations, the concept of the gradient is crucial. When you visualize this on a graph, the gradient describes how steep the line is. It is essentially the slope of the line and depicted as the letter \( m \) in equations. To calculate the gradient, you need two points on the line. By using these points' coordinates, \( (x_1, y_1) \) and \( (x_2, y_2) \), you can plug them into the formula:

• \(m = \frac{y_2 - y_1}{x_2 - x_1} \)

This ratio tells you the vertical change per unit of horizontal movement between the two points. A higher gradient means a steeper slope. For example, a gradient of 5 suggests that for every step you move horizontally, the line moves five steps vertically. This steepness is critical for predicting how quickly the line rises or falls as you move along the axis.

Understanding the y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. Imagine a graph where the line meets the y-axis; this meeting point corresponds to when \( x = 0 \). This point is denoted by the letter \( c \) in the linear equation.

To find the y-intercept in the equation \( y = mx + c \), once you've determined \( m \), you can substitute one of the known points into the equation to solve for \( c \). Using the given solution:

• Point \((1, -2) \)
• Gradient \( m = 5 \)

yields the calculation \(-2 = 5 \times 1 + c\), solving which gives \( c = -7 \).

This calculation shows where the line will intersect the y-axis once it is graphically displayed, helping to visualize the line's position on the graph.

Equation of a Line

The equation of a line, often written as \( y = mx + c \), is a straightforward representation of every point that lies on that line. In this equation, \( m \) represents the gradient, and \( c \) is the y-intercept.

Understanding this equation format helps in quickly determining how a line behaves just by looking at its formula. For instance, lines with the same gradient are parallel to each other, while those with different gradients will intersect. The y-intercept \( c \) determines how far up or down the line sits on the graph.

• For the given equation \( y = 5x - 7 \), the gradient is \( 5 \) and the y-intercept is \( -7 \).

This means the line rises steeply, increasing five units for every unit moved horizontally to the right, and intersects the y-axis at -7. Recognizing this interplay between the gradient and y-intercept helps in graphing the line and analyzing its behavior.