The Slope of a Straight Line.

In this section, we will delve into the concept of line gradients, and explore methods for calculating them through both graphical analysis and the use of coordinate pairs. Additionally, we will identify and discuss equations for horizontal and vertical lines.

Included within these resources are worksheets with questions derived from Edexcel, AQA, and OCR exams, as well as supplementary guidance for those requiring further assistance.

What exactly is a line gradient?

When it comes to measuring the angle of inclination of a straight line, the gradient, signified by the letter m in the general formula for linear equations, y = mx c, is the key element that determines the level of steepness.

Note that gradients can be either positive or negative, and are not limited to whole numbers. Depending on whether the line is travelling upwards ("uphill") or downwards ("downhill"), the gradient will be either positive or negative in value, respectively.

How does one calculate the gradient of a line?

Visualize a set of stairs with uniform steps. No matter how they're arranged, their height affects the speed at which you ascend. If each step is taller, you will finish a climb more quickly, while shorter steps will require more time.

The same principle applies to gradient comparisons. For instance, look at two sets of stairs. The blue set has steeper steps than the red set, resulting in a steeper gradient (indicated by the blue arrow being steeper than the red arrow). The height of the green steps is less than that of the red steps, causing the gradient to be gentler (as shown by the green arrow being shallower than the red arrow). Bear in mind that gradients are always determined from left to right and can be either positive or negative.

The rate of change between variables can be represented by the slope of a straight line graph, and the gradient of that line quantifies that rate. For example, the gradient can be instrumental in determining the conversion rate between various currencies.

Formula-wise, we utilize the gradient formula to determine the change in height, indicated by the y coordinates, divided by the change in width, represented by the x coordinates. If we have two coordinates, (x₁, y₁) and (x₂, y₂), then the gradients of straight lines can be computed using this formula:

• \[\m=\frac{y_-y_}{x_-x_}\]
• Lastly, this resource provides an overview of the equations governing horizontal and vertical lines.

The variation in the value of x is determined by the difference between the x coordinates, which is x₂ − x₁. In a similar manner, the difference between the y coordinates, y₂ − y₁, determines the shift in y. The formula for gradient is hence given as:

\[\m=\frac{y_-y_}{x_-x_}\]

The equation can also be perceived as 'Change in y divided by change in x' or 'Rise over run'.

The gradient equation is an alternative approach to representing the gradient or slope of a straight line using x and y coordinates, where the gradient equation is written as m = rise / run.

To determine the gradient of a line, divide the change in height (y₂ − y₁) by the change in horizontal distance (x₂ − x₁). For instance, consider a straight line with (4, 2) and (6, 8) as the points. The y coordinate difference is 6 (8 – 2), the x coordinate difference is 2 (6 – 4), and their ratio gives a gradient of 3.

Let's now examine the gradient of four lines. When m = 1, for each unit square moved to the right, one moves 1 unit square upwards. Similarly, when m = 2, one moves 2 unit squares upwards for each unit square moved to the right. When m = -3, for each unit square moved to the right, one moves 3 unit squares downwards. When m = 1/2, for each unit square moved to the right, one moves 1/2 a unit square upwards.

How much distance should separate the coordinates we select? Consider the case when m = 2. The gradient of the first blue line is 2, that of the second blue line is 2, and that of the third blue line is 2. The value derived will always simplify to m = 2, irrespective of how far apart the coordinates are on the line. Always use two coordinates that intersect the corners of two grid squares to determine the horizontal and vertical distance between them as accurately as possible. Try to use integers as much as possible. Remember that the change in x is horizontal, whereas the change in y is vertical.

Neither horizontal nor vertical lines have any bearing on the relationship between x and y, and hence it is impossible to express them in the form y = mx +c, since the gradient cannot be computed. Let's examine a few instances to expand our understanding of the equations of horizontal and vertical lines.

In the image above, all the coordinates share the x value of 4, regardless of the y value. Therefore, if one joins the coordinates to create a straight line, one gets the equation x = 4 for the vertical line.

Observe that the x-axis intercept of the line occurs at the point (4,0). It is worth noting that the y-coordinate of all the points in the diagram is 2, irrespective of their x-value. Consequently, joining these points together results in a horizontal line whose equation is y = 2. The line intercepts the y-axis at the point (0,2), denoting that the y-intercept is 2.

To calculate the gradient of a line, select two points that form the corners of two grid squares on the line. Next, sketch a right-angled triangle and label the corresponding change in y and change in x. Subsequently, divide the change in y by the change in x to obtain m, i.e., the gradient.

To obtain an estimate of the gradient of a curve at a specific point, draw a tangent line at that point, and calculate the gradient of the tangent line. As it touches the curve at just one point, the tangent line is an estimate drawn by eye, but should be as accurate as possible. Once the tangent is drawn in, use the method described above to calculate the gradient of the tangent line, which gives an estimation of the gradient of the curve at that point.

For instance, in the given example, the red curve represents the equation y=x^ and the blue tangent line was drawn to estimate the tangent at the point (2,4).

Do you want to assess your knowledge of gradient of a line? Then, download our free worksheet comprising 20 questions and answers that include reasoning and applied questions.

To determine the gradient of the line, begin by selecting two points on the line that fall on the corners of two grid squares. Next, draw a right angle triangle between those points and calculate the change in both the y and x coordinates. Keep in mind that the height of the triangle represents the change in y and the width represents the change in x.

For instance, take the coordinates (1, 2) and (5, -4). To move from (1, 2) to (5, -4), add 4 to the x coordinate and subtract 6 from the y coordinate. Since we want to calculate the change in y, we must express the height of the triangle as -6.

Finally, divide the change in y by the change in x to get the gradient, also known as m. In this case, \[-6/4 = -3/2\], so the gradient of the line is -3/2.

Identify two points on the line that fall at the corners of two grid squares.

Given this information, we can proceed to step 2.

Draw a right triangle and determine the difference in both the y and x coordinates.

Then, calculate the value of m by dividing the change in y by the change in x.

To find the slope of the line:

• First, choose two points on the line that lie at the corners of two grid squares.
• As we have coordinates A(4,3) and B(7,12) without a graph, we must ascertain whether the gradient is positive or negative, in addition to determining the value of m.
• Since there is no right angle to sketch, we will solve by calculating the change in y and the change in x algebraically.
• It's important to remember to subtract one coordinate from the other when finding the change in y and the change in x. For example, if we have coordinates A(3, 12) and B(4, 9), we would find:
• \[\y_-y_=9-12=-3\]
• \[\x_-x_=4-3=1\]
• To calculate the gradient of the line, we divide the change in y by the change in x to find m:
• \[\m=\frac{-3}\
• \]
• \[\m=-3\]
• If we don't have a graph to visualize the line, we can select two points on the line that occur on the corners of two grid squares. For example, to find the gradient of the line with coordinates P(−10, −3) and Q(2, −7), we solve algebraically by finding the change in y and the change in x:
• \[\y_-y_=-7- -3=-4\]
• \[\x_-x_=2- -10=12\]
• Then, we divide the change in y by the change in x to find m:
• \[\m=\frac{-4}\
• \]
• \[\m=-\frac\]

It's also important to keep track of the coordinates and not mix them up. Labeling each coordinate as (x₁, y₁) and (x₂, y₂) can help with this. If one or both of the numerator and denominator are negative when finding m, the gradient is negative. If both are negative, the gradient is positive.

It's easy to mistake the calculation of m to be the change in x divided by the change in y, but this would result in the reciprocal of the gradient which is usually incorrect. It's also important to read scales correctly, as they can change and affect the values. For example, if each square on the y-axis is 2 units, we need to account for this when finding the change in y.

Explore the links below to gain a better grasp of these algebraic equations:

• - Utilizing the equation \m=\frac\, we can calculate the slope of the given line.
• - By incorporating the equation \m=\frac{-9}\, we can obtain the slope of another line.
• - Another equation, \m=\frac=\frac\, when combined with the knowledge that each square unit on the x-axis is equal to two units, allows us to make further calculations.

The following information pertains to two straight-line graphs, A and B, displayed on the same set of axes. Line A's equation is represented by y=2(x 1).

2. The slope of line A can be calculated using the formula \m=\frac{-3}. Here, each square on the x-axis is equal to \frac unit.

3. The slope of line B can be calculated using the formula \m=\frac{24-6}{8-2}=\frac=3.

4. Line B's slope can also be calculated alternatively using the formula \m=\frac{10–8}{-5–3}=\frac{10 8}{-5 3}=\frac{-2}=-9.

Find the gradient of line B using the following analysis.

(4 marks)

Show answer

The y-intercept of line A is labeled as 2 or (0,2).

(1)

Each square on the x-axis measures one unit and on the y-axis measures two units.

(1)

Two coordinates must be selected on line B, with the correct change in y and change in x, for example:

• The change in y is -8 and the change in x is 4.
• (1)
• \m=\frac{y_-y_}{x_-x_}=\frac{-8}=-2\
• (1)
• Write an equation in the form y=mx+c that has the same gradient as the line 3y-9=12x.
• Tick the statements that are true for the equation 3y-9=12x and for your solution for part a).
• They are parallel.
• They intersect at the coordinate (0,-3).
• They are perpendicular to one another.
• (4 marks)
• Show answer
• (a)
• (1)
• m=4.
• (1)
• Any equation of the form y=4x+c.
• For example, y=4x, y=4x+7, y=4x-8.
• (1)
• (b)
• They are parallel ✔.
• They intersect at the coordinate (0,-3).
• They are perpendicular to one another.
• (1)
• Craig is calculating the gradient of the following straight line.

Is Craig's calculation correct? Explain your answer.

(3 marks)

Show answer

No (with reasoning).

(1)

The change in x is equal to:

• -6–2=-6 2=-4
• whereas Craig has stated the height of the triangle as 4.
• (1)
• The gradient is
• \m=\frac{y_-y_}{x_-x_}=\frac{-4}=-\frac
• whereas Craig has incorrectly calculated
• \m=\frac{x_-x_}{y_-y_}
• You have now learned how to:
• Calculate and interpret gradients of linear equations' graphs numerically, graphically, and algebraically.
• Interpret the gradient of a line graph as a rate of change.
• Maths formulas, interpreting graphs, and inequalities are helpful resources for GCSE maths success.

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