What is a family of lines?

Just think about your family members. You most likely have a lot in common, yet you are not entirely the same. The same may be true for the line family. What characteristics does a line family share? What may be different?  Many mathematicians and experts have evidenced that a straight line has two essential properties: slope and y-intercept. The inclination indicates how steeply the straight line climbs or falls, while the y-intercept indicates where the line crosses the y-axis. We can demonstrate this concept with two families of straight lines to illustrate this notion. A “family of straight lines” is a collection of unbroken lines that share a similar feature. 

We can classify straight lines into two groups: those with the same slope and y-intercept location and those without it. 

We can say that we have two types of family lines. 

• Keep the slope constant while varying the y-intercept; parallel lines
• Change the slope while keeping the y-intercept constant.

What is the general equation of the family of lines?

The universal equation of the family of lines via the point of intersection of two provided lines is L + λ L ‘= 0, where L = 0 and L’ = 0 are the two assigned lines, and λ works as a parameter.

An explanation and understanding of the general equation of the family of lines are as mentioned below. 

• It forms a line of the form L =  L1 + λ L2 = 0. On the other hand, it goes through the point formed by the junction of the lines L1 = 0 and  L2 = 0. (where λ is a parameter) .
• A linear equation with an indeterminate coefficient can depict a family of straight lines. It is necessary to note that the straight line L = 0 cannot be a fixed line in this scenario.
• We understand that L1 and L2 meet at infinity if they are parallel.
• If ax + by + c = 0, the perpendicular line is bx-ay + k = 0, where k is a parameter.
• If we have a line ax + by + c = 0, then ax + by + k = 0, where k is a parameter, is the line parallel to it.

We can define the equation of a line as having a slope and an intercept form.

The equation y = mx + b may describe a straight line in the coordinate plane, where m is the slope and b is the intercept. 

How will you find the equation of the family of lines with an x-intercept -6?

Solution

• The family’s x-intercept is set to -6, with each member passing through the point (-6, 0).
• The equation of such a family of lines in point-slope form is y-0 = m (x-(-6)), i.e., y = m (x + 6), where m is a parameter. 
• The above equation of the family does not give the vertical line through the point (-6, 0). 
• However, the equation of this line is x = -6, i.e., x + 6 = 0.

The equation of a straight line is sometimes given to us in a different form.

How can we show that this represents a straight line and find its gradient and intercept value on the y-axis?

• Suppose we have the equation 5y – 5x = 25.
• We can use algebraic rearrangement to obtain an equation in the form of y = mx + c: 
• 5y-5x = 25 ⇨   5y = 25+5x  ⇨ y =5x+255  or y = x+5
• So now the equation is in its standard form, and we can see that the gradient is 1, and the intercept value on the y-axis is +5.

We can also work in a reverse way. Suppose we know that a line has a gradient of 1/5 and has a vertical intercept at y = 3.

What would its equation be? 

• To find the equation, we substitute the correct values into the general formula y = mx + c. Here, m is 1/5, and c is 3, so the equation is y = x5 + 3. If we want to remove the fraction, we can also give the equation y =  x+155 

How do you solve the equation for a straight line with a 3 unit x-intercept parallel to 4x-7 y = 11?

Solution

According to the equation, the presented line is 4x – 7y – 11 = 0…. …..(1)

As per the above equation of the family of lines parallel to

• (1) 4 x -7 y+k = 0…… …. …………(equation number 2 will be)….(2)
• (2), where k denotes the parameter

By putting y = 0, you can find the x-intercept.

• We get 4 x + k = 0 => x = – k4
• For the required members of the family who make x-intercept 3 units,
• – k4 = 3, which equals k = -12.
• We can replace this value of k in (2), the equation of the required line is
• 4 x – 7 y – 12 = 0

How will you write an equation of the family of lines satisfying the following conditions?

1. parallel to the x-axis
2. through the point (0, -1)

1)  All lines with a zero gradient are parallel to the x-axis, and the y-intercept can be anything. In conclusion, the straight line family is y = 0x+b, abbreviated as y = b.

2) All lines travelling through the point (0,-1) have the same y-intercept and b = -1 according to the given criteria.

As a result, a line family becomes y = mx -1.

How do you investigate families of lines?

Family 1:  Keep the slope unchanged and vary the y−intercept.

The  family of lines of the equations is of the form y=−2x+z.Although all the lines have a slope of –2, the value of z varies between them. We may note here that all the lines in this family are parallel. All lines are identical except that they shift up and down the y-axis. The line rises on the y-axis as b increases, and as b decreases, the line falls on the y-axis. We describe it as a “vertical shift”  type of behaviour. 

Family 2:  We can change the slope while keeping the y-intercept constant. The family of lines with equations of the form y=mx+2 is considered. The slope varies, even though all the lines have a y-intercept of two. The higher the value of “m,” the sharper the line.

Conclusion 

With this, we have gained a good amount of knowledge and got an insight into the topic ‘ Family of straight lines ’. Here we have discussed the representation of a family of lines and have solved several examples on the same. Family of straight lines is a very important topic, it helps in understanding some more advanced topics in coordinate geometry to follow.