Linear Equation

An equation is called linear when the variable’s highest power is 1. This means you won’t find squared, cubed, or high-power variables (like \({{x^{2}}, {x^{3}}…}\)) in the equation. It includes variables raised only to the first power and does not involve any products of variables. The values change at a constant rate, which makes the graph of the linear equation a straight line. The graph of a linear equation is always a straight line.

Example:

\(2x + 6 = 10\)

\(x = 2y\) (This shows a linear equation with two variables).

\(x + y = 5\)

Facts About Linear Equations

• The linear equation always produces a graph that forms a straight line, with no curves or bends.

• The variable’s power is always 1.

• Linear equations can have one, two, or more variables.

• A linear equation remains balanced as long as you add, subtract, multiply, or divide both sides by the same value. 

• It is also known as a one-degree equation.

• The graph ends with a straight-line pattern. 
   

Think of an equation like a seesaw, both sides must stay balanced. In math, an equation means the left side is equal to the right side. Whatever you do on one side of the equation, do the same on the other side; otherwise, it will be unequal. We move the variable to one side and the constant to the other to simplify the equation for easier solving.  

When solving an equation, you can isolate the variable by moving numbers to the other side. Performing the opposite operation changes the sign accordingly. (\({2x + 3 = 7 }\) becomes \({2x = 7 - 3}\)). The +3 became -3 because you're using the opposite operation to cancel it out.

Let’s see the example,

\({{{5x} - 8 = 7}}\)

Here, we don’t know the value of x. So we move the constant from LHS to RHS, it becomes

\({{{5x} = 7 + 8}}\)

Then add the constants 

\({5x = 15}\)

Both 5 and 15 can be divided by 5, so we divide both sides of the equation by 5.

Dividing by 5: \({{{5x\over 15} = {15\over5}}}\)
 

Therefore, the value of x is 3.
 

A linear equation is an algebraic equation that involves variables, such as x and y, and constants, including numbers or fixed values. In a linear equation, the highest power of the variable is 1. There are several forms, namely the standard form, slope point form, slope-intercept form, etc. For example, \({{{3x - 2y} = {12}}}\) (3, 2, and 12 are the constants, and x, y are the variables).

The Standard Form: The standard form of a linear equation includes both constants and variables. In the standard form, there are different types of linear equations—some have just one variable, while others include two.

One Variable Form (Simple) : In one variable form, an equation contains one variable. For e.g., \({{{Ax + B} = {0}}}\)
In this equation, A and B are constants, and x is the variable. 

Two Variable Form (Standard Form)

\({{{Ax + By + C} = {0}}}\)

A, B, and C are constants 

x and y are variables

Both A and B cannot be 0 simultaneously.

Slope Point Form: The slope indicates the line’s steepness and direction. If a line has a slope m and passes through the point \({(x1, y1)}\), we can write its equation in slope-point form.
\({{{{y − y_{₁}}} = {{m(x − {x_{₁}})}}}}\). You can use this form when you already know the slope of the line and one point it passes through.

Slope-Intercept Form: A common and easy way to write a linear equation is \({{{{(y}} = {{mx + B)}}}}\), where (x) and (y) are the variables, m is the slope of the line, and B is the intercept (Intercept means the point where the line crosses the y-axis)

Example:

\({{{{y}} = {{3x + 1}}}}\) 

Slope m = 3

y-intercept b = 1
 

The standard form is also known as the two-variable form. The equation contains two variables. In the Standard form, there are two different variables contained in the same equation.

\({{{Ax + By} = {C}}}\)

A, B, and C are constants

x and y are variables

Example:

\({{{2x + 3y} = {12}}}\)
 

The linear equation graph is the solution that can visually show the straight line, which is why it is called linear. It shows the relationship between the x-axis and the y-axis. The line depends on the slope. When a line crosses the y-axis, that point is called the y-intercept. 
 

Example:

\({{{y} = {2x + 1}}}\)

Take the value of x as \({{(-1, 0, 1)}}\)

To find the value of y: 

If x is -1, \({{{y} = {2{(-1)}} + 1 \implies {y = -1}}}\)

If x is 0, \({{{{y} = {2{(0)}} + 1 \implies {y = 1}}}}\)

If x is 1, \({{{{y} = {2{(1)}} + 1 \implies {y = -3}}}}\)

The points are \({{({-1},{-1}), (0,1), (1,3)}}\)
 

A linear equation with one variable uses that variable consistently throughout the equation. 

\({{Ax + B = 0}}\)

\({{Ax + Bx + C = 0}}\)

Example

\({{5x = 20}}\)

\({{x = 4}}\)

\({{{{5{(4)}} = {20}}}}\)

The value of x is 4
 

In the equation, there are two different variables. Both variables have a degree of 1. For example, the equation \({{(5x + 2y + 8 = 0)}}\) has variables (x) and (y), and the numbers 5, 2, and 8 are constants.
This equation contains a straight line.

For example

\({{2x + 3y = 12}}\)

In this equation, we want to find the value of x and y

Let’s take the value of x is 0

then \({{2(0) + 3y = 12}}\)

 \({{3y = 12}}\)

 \({{{{y}\over3} = {12\over 3} }}\) (The 3 is multiplying the variable on the left-hand side (LHS). To isolate y, we divide both sides by 3.)

\({{y = 4}}\)

The value of x and y is (0,4).
 

Think of a linear equation as a scale: both sides must stay the same to stay balanced. 
We can do the same thing on both sides (LHS, RHS) so the balance is not disturbed.

Step 1: Combine the equations in simplified form for better understanding

Step 2: Change the variable from one side to the other side of the equation

Step 3: Solve the equation

Example: 

Solve the Linear equation \({{{3x} - {2} = {4}}}\)

Solution: 

Change the constant LHS to RHS

\({{{3x} - {2} = {4}}}\)

Add 2 on both sides of the equation

\({3x} - {2} + {2} = {4} + {2}\)

\({{3x} = {6}}\)


When the value 3 is moved from the left-hand side (LHS) to the right-hand side(RHS), it becomes a divisor because we’re dividing both sides by 3. 

\({{x} = {{6} \over {2}}} \)

\(x = 2\).

Learning linear equations becomes easier when students understand the simple pattern. In this section, we will learn some tips and tricks to master linear equations. 

• Understand the basic concept: Linear equations are equations where the highest power of the variable is 1. For example, \({{2x + 3 = 9}}\).
   
• Keep the balance: When solving linear equations, whatever you do to one side, do the same to the other side. To keep the equation true, when adding, subtracting, multiplying, or dividing, do it on both sides. 
   
• Combining the like terms: Simplify the linear equation by adding or subtracting similar terms. For example, in \({{5x + 8 + 2x = 50}}\), it can be combined to get \({{{{7x + 8 = 50}}}}\).
   
• Isolating the variable: To isolate x alone, first move the constant and then move the multiplier. For example, solving \({{{{5x + 2} = {12}} }}\)
  
  \({{\implies {{5x} ={12 - 2}} }}\)
  
  \({{\implies {{5x} = {10}}}} \)
  
  \({{{\implies {{x}} = {{10 \over 5}}}}}\)
  
  \({{\implies {{x} = {2}}}}\)
   
• Visual representation: When solving linear equations, students can use a small table to mark the given values and solve the equation. This helps students to visualize how the equation behaves. 

Linear equations are used to solve everyday problems in many fields such as planning, calculating, and decision-making. In this section, we will learn a few real-life applications of linear equations.

• In budgeting and saving, linear equations are used to plan how to save money. For example, if Ria wants to buy a LEGO set costing \({{$1000}}\), and he saved \({{$200}}\) and earns \({{$50}}\) per day. Let x be the number of days he works: \({{{50x + 200 = 1000 \implies 50x = 800 \implies x = 16}}}\). So Riya needs to work 16 days to buy the LEGO set.
   
• Linear equation is used in cooking and recipes to adjust recipes for any number of servings. For example, if 2 cups of rice serve 4 people, then for x people, the rice needed is 0.5x.
   
• In traveling and time management, a linear equation is used to calculate distance, time, or speed. For example, driving at 60 km/h for x hours gives a distance \({{= 60 \times x}}\). To travel 180 km, \({{{60x = 6 × 180 = 1080 {\text { min}} = 3 {\text { hours}}}}}\).
   
• While shopping, we use linear equations to calculate the costs after discounts. For example, if a jacket costs \({{$200}}\) with a \({{$20}}\) discount per jacket. The total cost for x jacket is \({{{{200x} - {20x} = {180x}}}}\). So buying 5 jackets will cost is: \({{{{180} \times {5} = {900}}}}\).
   
• To monitor and plan mobile data usage, we use linear equations. For example, if your plan gives 2 GB per day, and you have 5 GB left, the total data used after x days is \({{{{2x} + {5}}}}\). For 15 Gb limit, \({{{{2x} + {5} = {15}} \implies {{x} = {5 \text {days}}}}}\).