Linear equations are used in various different fields for solving real-world problems. In this post, we’ll take a look at what linear equations in one variable are and how they can be solved.
Linear equations are simply an equation with one or more variables that have only one unknown value – the “solution” for that variable. In order to solve this type of equation, you either need to use the technique of substitution or mathematical elimination. Let us look at some examples which will give more clarity on – what are linear equations?
Example 1: Find the answer to the equation 4x – 1 = 7
Solution:
- Step 1: Add 1 to both sides. 4x – 1 + 1 = 7 + 1 (balance is not disturbed)
- Step 2: Divide both sides by 4 so, 4 x/4 = 8/ 4 or x = 2 (required solution)
Example 2: Solve 13x – 26 =39
Solution:
The first thing to be noticed in this equation is that it is not in its simplest form; for example, we can see that every term in the equation has a multiple of 13. This implies that the equation can be written in the form,
13 (x – 2) = 13 * 3
As both RHS and LHS have multiples of 13 so we can divide both sides by 13 and the equation reduced to its simplest form,
X – 2 = 3
X = 5
Equations are very useful in day-to-day applications and are not just a theoretical concept; let us see below some examples to appreciate the utility of linear equations, how they are formed, and how they are solved to get the value of the unknown
Example 1: Suresh was creating a swimming pool that was rectangular in shape and had a perimeter of 100 m, and the width of the swimming pool was 40 m. Now, Suresh needs to tell the construction team what should be the length of the swimming pool. Can you help Suresh?
Now, the two values known in this puzzle are the perimeter of the swimming pool, which is 100 mts, and the width of the swimming pool, which is 40 m.
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The unknown is the length of the swimming pool.
Let us now list down some of the relations we know because of the shape of the swimming pool, which is rectangular.
We know that the perimeter of a rectangle is twice its length + width.
So, perimeter 100 is equal to Length which is unknown and width which is 40 mts.
To reduce this long sentence, we use some symbols :
Length of the swimming pool = x
Perimeter = 2 ( x + 40 )
100 = 2 ( x + 40 )
50 = x + 40
50 – 40 = x
10 = x
So, the length of the swimming pool is 10 meters.
Example 2: Ram is at present 15 years old, and his father Kevin is twice as old as Ram’s sister Mira’s, who was 4 years older than Ram five years back. So, can Ram calculate the age of his father Kevin ?
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Let us again see the known facts and unknown variables separately.
Known facts – The age of Ram is 15 years at present, also known is that Kevin was two times as old as Mira and Mira is five years older than Ram.
Please see that while the problem states that Mira was five years older than Ram a few years back, we can reason that the difference in the age remains the same over the years.
So, Let’s say Kevin’s age is = x years
Then, Mira’s age is 15 + 4 = 19 years
And as Kevin is twice as old as Mira, so x = 2 * 19 = 38
So, Kevin’s age is 38 years.
Students can download the linear equation worksheets from the Cuemath website to practice more on such problems. These are free to download and can be printed too for home or classroom use. These worksheets are also provided with the answer keys having detailed step-by-step solutions, which will help students understand the problem-solving techniques much better.
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