Linear Equations

• Linear Equations
• What is a Linear Equation?
• Different Forms of Linear Equations
• Examples of Non-Linear Equations
• How to make Graph Linear Equations?
• How to Solve Linear Equations?
• Real-Life Applications of the Linear Equations
• Solved Examples
• Practice Problems
• Frequently Asked Questions

Linear Equations

A linear equation is a fundamental concept in algebra that represents a straight line on a graph.

It's an equation where each term is either a constant or the product of a constant and a single variable raised to the first power.

In other words, it's an equation that can be graphically represented as a straight line. 

Linear equations are widely used in various fields such as mathematics, physics, engineering, economics, and more, for modeling and solving real-world problems.

What is a Linear Equation?

It is an algebraic equation of the first degree, meaning that the highest exponent of the variable is `1`. 

The key characteristics of linear equations are that the variables are raised to the power of `1`, and when graphed, the equation represents a straight line. 

The general form of a linear equation in one variable is

 \( ax + b = 0 \),

 where \( a \) and \( b \) are constants, and \( x \) is the variable. 

Example: 

\( 2x - 3 = 0 \)

This equation represents a straight line when graphed, and its solution is  \(x = \frac{3}{2}\).

In two variables, the general form is:

\( ax + by + c = 0 \)

where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables.

Example:

\( 3x + 2y = 6 \)

This equation represents a line in the `xy`-plane, and its solutions are pairs of values \((x, y)\) that satisfy the equation, such as \(x = 2\) and \(y = 0\). We can say that the point `(2, 0)` satisfies the equation, meaning the point `(2, 0)` lies on the line \( 3x + 2y = 6 \).

Different Forms of Linear Equations

Linear equations can be represented in various forms, each offering different insights and advantages. Here are some common forms of linear equations with examples:

`1`. Slope-Intercept Form:

   \( y = mx + b \)

   Example: \( y = 2x + 3 \)

   The slope of the line is \(m = 2\) and the `y`-intercept is \(b = 3\).

`2`. Point-Slope Form:

   \( y - y_1 = m(x - x_1) \)

   Example: \( y - 4 = 2(x - 1) \)

   This equation represents a line with slope \(m = 2\) passing through the point \((1, 4)\).

`3`. Standard Form:

   \( Ax + By = C \)

   Example: \( 2x + 3y = 6 \)

Here `A,B` and `C` do not directly give any feature of the graph like slope, intercepts, etc. However we can calculate slope as `-A/B` and `y`-intercept as `C/B`.

Each form provides a different perspective on the linear equation and are useful in different situations.

Examples of Non-Linear Equations

To help us identify linear equations better, let us look at some examples of equations that are not linear. Below are examples of not linear equations:

`1`. Quadratic Equation:

   \( 3x^2 + 2x - 5 = 0 \)

   This is a quadratic equation because the highest power of the variable (\(x\)) is `2`.

`2`. Square Root:

   \( \sqrt{x} + 4 = 2 \)

   The square root of \(x\) makes it a non-linear equation.

`3`. Fractional Exponent:

   \( 2x^{\frac{1}{3}} - 3 = 0 \)

   The fractional exponent `(1/3)` makes it a non-linear equation.

`4`. Absolute Value:

   \( |2x - 1| = 3 \)

   The absolute value function makes this equation non-linear.

`5`. Trigonometric Equation:

   \( \sin(x) + 2 = 0 \)

   Involving trigonometric functions makes this non-linear.

In these examples, the variables are raised to powers other than `1`, and involve square roots, absolute values, or trigonometric functions, making them non-linear equations. Linear equations specifically represent straight-line relationships, and any deviation from a first-degree polynomial makes an equation non-linear.

How to Graph Linear Equations?

The graph of a linear equation is a straight line when plotted on a coordinate plane. The equation \(y = mx + b\) is often used to understand the graph of a linear equation. Here are some key aspects related to linear graphs:

`1`. Slope (\(m\)): The slope is the coefficient of \(x\) in the equation \(y = mx + b\). It determines the steepness or direction of the line. A positive slope indicates a rising line, while a negative slope indicates a falling line. A slope of zero results in a horizontal line. A vertical line has an undefined slope.

`2`. `Y`-Intercept (\(b\)): The `y`-intercept is the point where the line intersects the `y`-axis. It is the value of \(y\) when \(x = 0\). In \(y = mx + b\), `b` represents the `y`-intercept.

Steps to graph a linear equation:

1. Identify the slope (\(m\)) and the `y`-intercept (\(b\)).
2. Plot the `y`-intercept on the `y`-axis.
3. Use the slope to find additional points on the line. For example, if the slope is \(2\), move up by \(2\) units vertically and \(1\) unit horizontally from the y-intercept to find another point.
4. Connect the points to draw the straight line.

Example:  Graph the linear equation: \( y = 2x + 3 \).

`1`. Identify the slope and `y`-Intercept:

• Slope (\(m\)) `= 2`
• `Y`-Intercept (\(b\)) `= 3`

`2`. Plot the `Y`-Intercept:

• Plot the point `(0, 3)` on the `y`-axis.

`3`. Use the Slope to Find Another Point:

• We can write the slope `2` as \(\frac{2}{1} \), indicating \(\frac{\text{rise}}{\text{run}} \). 
• Continue from the `y`-intercept, move up `2` units and to the right `1` unit to find the next point `(1, 5)`.

`4`. Connect the Points:

• Draw a straight line through the two plotted points.

How to Solve Linear Equations in One Variable?

Solving a linear equation in one variable involves isolating the variable on one side of the equation. Here are the general steps to solve a linear equation in one variable (\(x\)):

`1`. Start with the equation:

   Begin with the given linear equation. For example:

   \( 2x + 5 = 11 \)

`2`. Isolate the variable term:

   Use inverse arithmetic operations to isolate the \(x\)-term.

   \( 2x + 5 - 5 = 11 - 5 \)

   This simplifies to \(2x = 6\).

`3`. Isolate the Variable:

   Divide both sides of the equation by the coefficient of the variable to isolate \(x\).

   \( \frac{2x}{2} = \frac{6}{2} \)

   This gives \(x = 3\).

`4`. Check the Solution:

   Substitute the found value back into the original equation to verify that it satisfies the equation.

   \( 2(3) + 5 = 11 \)

   This is true, confirming that \(x = 3\) is the solution.

So, the solution to the equation \(2x + 5 = 11\) is \(x = 3\).

Remember, these steps apply to linear equations in one variable. For more complex equations or systems of equations, additional methods like substitution, elimination, or matrices may be needed.

Real-Life Applications of the Linear Equations

Linear equations have numerous real-life applications across various fields. Here are a few examples:

1. Finance: Budgeting and financial planning often involve linear equations. For instance, you might model your monthly expenses and income using linear equations to determine if you are on track with your budget.
    
2. Physics: In physics, linear equations are used to describe the motion of objects. The equation \(s = ut + \frac{1}{2}at^2\) describes the position (\(s\)) of an object in terms of initial velocity (\(u\)), time (\(t\)), and acceleration (\(a\)).
    
3. Economics: Economic models often use linear equations to represent relationships between variables. For example, the demand and supply curves can be modeled using linear equations.
    
4. Engineering: Engineers use linear equations in designing structures and systems. For instance, in civil engineering, the load-bearing capacity of a beam can be modeled using linear equations.
    
5. Computer Science: Linear equations are used in computer graphics for tasks such as rendering and animation. They help determine the position and movement of objects on the screen.
    
6. Environmental Science: Linear equations can model ecological relationships, such as population growth or decay. They can help predict the future size of a population based on current trends.
    
7. Health Sciences: Linear equations can be used in medicine to model drug dosage or the concentration of substances in the body over time.
    
8. Business and Marketing: Companies use linear equations for pricing strategies, revenue projections, and sales forecasting. For example, the total revenue (\(R\)) can be modeled as \(R = px\), where \(p\) is the price per unit and \(x\) is the quantity sold.
    
9. Telecommunications: Linear equations are used in telecommunications to model signal strength, network traffic, and data transfer rates.

Solved Examples

Example `1`. Solve the equation `2x+3=7`.

Solution:

Step `1`: Subtract `3` from both sides

\( 2x + 3 - 3 = 7 - 3 \)

This simplifies to:

\( 2x = 4 \)

Step `2`: Divide both sides by `2`

\( \frac{2x}{2} = \frac{4}{2} \)

This gives:

\( x = 2 \)

So, the solution to the equation \(2x + 3 = 7\) is \(x = 2\). You can check this by substituting \(x = 2\) back into the original equation:

\( 2(2) + 3 = 4 + 3 = 7 \)

Therefore, \(x = 2\) satisfies the equation.

Example `2`. Solve the equation `3x−2(x−1)=5`.

Solution:

Step `1`: Distribute the `-2` on the left side:

\(3x - 2x + 2 = 5\)

Combine like terms:

\(x + 2 = 5\)

Step `2`: Subtract `2` from both sides:

\(x + 2 - 2 = 5 - 2\)

This simplifies to:

\(x = 3\)

So, the solution to the equation \(3x - 2(x - 1) = 5\) is \(x = 3\). You can check this solution by substituting \(x = 3\) back into the original equation:

\(3(3) - 2(3 - 1) = 5\)

\(9 - 2(2) = 5\)

\(9 - 4 = 5\)

\(5 = 5\)

Therefore, \(x = 3\) satisfies the equation.

Example `3`. Solve the equation `2(2y−1)=6`.

Solution:

Step `1`: Distribute the `2` on the left side:

\( 4y - 2 = 6 \)

Step `2`: Add `2` to both sides:

\( 4y - 2 + 2 = 6 + 2 \)

This simplifies to:

\( 4y = 8 \)

Step `3`: Divide both sides by `4`:

\( \frac{4y}{4} = \frac{8}{4} \)

This gives:

\( y = 2 \)

So, the solution to the equation \(2(2y - 1) = 6\) is \(y = 2\). You can check this solution by substituting \(y = 2\) back into the original equation to ensure both sides are equal:

\( 2(2(2) - 1) = 6 \)

\( 2(4 - 1) = 6 \)

\( 2(3) = 6 \)

\( 6 = 6 \)

Therefore, \(y = 2\) is the correct solution.

Example `4`. Solve the equation \(\frac{2}{3}x - 5 = 1\).

Solution:

Step `1`: Add `5` to both sides:

\( \frac{2}{3}x - 5 + 5 = 1 + 5 \)

This simplifies to:

\( \frac{2}{3}x = 6 \)

Step `2`: Multiply both sides by the reciprocal of \(\frac{2}{3}\), which is \(\frac{3}{2}\):

\( \frac{3}{2} \cdot \frac{2}{3}x = \frac{3}{2} \cdot 6 \)

This simplifies to:

\( x = 9 \)

So, the solution to the equation \(\frac{2}{3}x - 5 = 1\) is \(x = 9\). You can check this solution by substituting \(x = 9\) back into the original equation to ensure both sides are equal:

\( \frac{2}{3}(9) - 5 = 1 \)

\( 6 - 5 = 1 \)

\( 1 = 1 \)

Therefore, \(x = 9\) is the correct solution.

Example `5`. Solve \(10 - \frac{3}{2}y - 2y = 6\), for `y`.

Solution:

Step `1`: Combine like terms on the left side:

\( 10 - \frac{3}{2}y - 2y = 6 \)

\( 10 - \frac{7}{2}y = 6 \)

Step `2`: Subtract `10` from both sides:

\( 10 - 10 - \frac{7}{2}y = 6 - 10 \)

This simplifies to:

\( -\frac{7}{2}y = -4 \)

Step `3`:  Multiply both sides by the reciprocal of \(-\frac{7}{2}\), which is \(-\frac{2}{7}\):

\( -\frac{2}{7} \cdot \left(-\frac{7}{2}y\right) = -\frac{2}{7} \cdot (-4) \)

This simplifies to:

\( y = \frac{8}{7} \)

So, the solution to the equation \(10 - \frac{3}{2}y - 2y = 6\) is \(y = \frac{8}{7}\). You can check this solution by substituting \(y = \frac{8}{7}\) back into the original equation to ensure both sides are equal:

\(10 - \frac{3}{2}(\frac{8}{7}) - 2(\frac{8}{7}) = 6\)

\(10 - \frac{12}{7} - \frac{16}{7} = 6\)

\(10 - \frac{28}{7} = 6\)

\(10 - 4 = 6\)

\(6 = 6\)

Therefore, \(y = \frac{8}{7}\) is the correct solution.

Practice Problems

Q`1`: Solve the equation `2x - 3 = 7`.

1. `x=8`
2. `x=` \( \frac{3}{2} \)
3. `x=4`
4. `x=5`

Answer: d

Q`2`: Solve the equation `2(x - 2) - 5(x - 4) =   20`

1. `x=`\(- \frac{3}{4} \)
2. `x=`\(\frac{3}{4} \)
3. `x=`\(- \frac{4}{3} \)
4. `x=`\(\frac{4}{3} \)

Answer: c

Q`3`: Solve the equation `72y-7 = 28y -5`

1. `y =` \( \frac{22}{1} \)
2. `y =` \( \frac{1}{22} \)
3. `y =` \( \frac{4}{5} \)
4. `y = 8`

Answer: b

Q`4`: Solve the equation \( \frac{{8y - 5}}{{7y + 1}} = -\frac{4}{5} \)

1. `y=`\( \frac{21}{68} \)
2. `y=`\( \frac{21}{60} \)
3. `y=48`
4. `y=36`

 Answer: a

Q`5`: Solve the equation `49y + 3 = 10`

1. `y=`\( \frac{3}{2} \)
2. `y=`\( \frac{7}{2} \)
3. `y=`\( \frac{1}{7} \)
4. `y=8`

Answer: c

Frequently Asked Questions

Q`1`. What is a linear equation?

Answer: A linear equation is an algebraic equation of the first degree, meaning the highest exponent of the variable is `1`. It represents a straight line when graphed.

Q`2`. What is the standard form of a linear equation?

Answer: The standard form of a linear equation in two variables (\(x\) and \(y\)) is \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants, and \(a\) and \(b\) are not both zero.

Q`3`. How do you solve a linear equation?

Answer: To solve a linear equation, isolate the variable on one side of the equation by performing inverse operations (addition, subtraction, multiplication, division) on both sides.

Q`4`. What are the methods for solving a system of linear equations?

Answer: Common methods include substitution, elimination, and matrix methods. These techniques involve manipulating the equations to find the values of the variables that satisfy all the equations simultaneously.

Q`5`. What is the slope-intercept form of a linear equation?

Answer: The slope-intercept form is \(y = mx + b\), where \(m\) is the slope of the line, and \(b\) is the `y`-intercept (the point where the line crosses the `y`-axis).

Q`6`. What is the point-slope form of a linear equation?

Answer: The point-slope form is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line, and \(m\) is the slope.

Q`7`. How do you graph a linear equation?

Answer: Plot at least two points on the coordinate plane and connect them with a straight line. The slope-intercept form \(y = mx + b\) is often used for graphing where the y-intercept is the preferred first point to be plotted.

Q`8`. What does a system of linear equations represent geometrically?

Answer: A system of linear equations represents multiple lines in the coordinate plane. The solution to the system is the point where all the lines intersect.

Q`9`. Can linear equations have no solution or infinitely many solutions?

Answer: Yes, a system of linear equations can have no solution (inconsistent system) or infinitely many solutions (dependent system) depending on the relationships between the lines.

Q`10`. What is the role of linear equations in real-life applications?

Answer: Linear equations are used to model and solve problems in various fields, including physics, engineering, economics, and computer science. They describe relationships that exhibit a constant rate of change.