# Linear equations

**What are linear equations?**

A linear equation can be written in different ways. An equation containing two variables $$x$$ and $$y$$ is said to form a linear equation in two variables. The highest degree of both the variables $$x$$ and $$y$$ in the equation is $$1$$.

Linear equations are of two types, Linear equation in one variable and Linear equation in two variables. Here we discuss the types of linear equations.

**Linear equation in one variable**

A linear equation in one variable is an equation written in the form of $$ax+b=0$$, where $$a$$ and $$b$$ are the real numbers in which $$a$$ is said to be the coefficient of variable of the given equation and $$b$$ is said to be the constant of the given equation.

**Linear equation in two variables**

A linear equation in two variables is an equation which can be written as in the form of $$ax+by+c=0$$, where $$a$$ and $$b$$ are not both equal to $$0$$. Here $$a$$, $$b$$, $$c$$ are the real numbers. The numbers $$a$$, $$b$$ are the coefficients of the variables of the given linear equation, and $$c$$ is called the constant of the given linear equation.

**E2.5: Derive and solve the linear equation in one and two unknown variables**

The concepts of linear equations are used in various real-life problems, like problems based on age, based on speed, distance and time, based on money and percentage, etc.

*Example 1*

Suppose Shezan has bought $$7$$ Arabian jasmine flowers at $$\$42$$. Let us construct the equation and calculate the price of $$1$$ Arabian jasmine flowers.

Let the price of one flower be $$\$x$$.

Then the price of $$7$$ flowers be $$\$7x$$.

Therefore, by the given problem, $$7x=42$$.

Divide both the sides of the equation by $$7$$ to get $$x=6$$.

Therefore, the price of $$1$$ Arabian jasmine flower is $$\$6$$.

The equation $$7x=42$$ represent a linear equation in one variable.

*Example 2:*

Rachel has bought $$1$$ balloon and $$2$$ chocolates at $$\$15$$ but Mahi bought $$1$$ balloon and $$3$$ chocolates at $$\$20$$. Construct the simultaneous equations for these two conditions.

Let us suppose, price of one balloon be $$x$$ and the price of one chocolate be $$y$$.

Therefore, the price of one balloon be $$\$x$$ and the price of two chocolates be $$2y$$.

Then, by the given condition, the first equation for the cost of the balloons and the chocolates is $$x+2y=15$$.

Now, the price of one balloon be $$\$x$$ and the price of three chocolates be $$3y$$.

Then, by the given condition, the second equation for the cost of the balloons and the chocolates is $$x+3y=20$$.

Therefore, the equations $$x+2y=15$$ and $$x+3y=20$$ are linear equations in the two variables.

**Worked examples of linear equations:**

**Example 1:** Find the value of $$x$$ for the equation $$ 5x-3=3x+7$$.

**Step 1: Subtract $$3x$$ from both the sides of the equation. **

$$2x-3=7$$

**Step 2: Subtract $$7$$ from both the sides of the equation.**

$$2x-10=0$$

**Step 3: Add $$10$$ to both the sides of the equation.**

$$2x=10$$

**Step 4: Divide** **both sides by $$2$$ of the given equation.**

$$x=5$$

**Step 5: Write the final answer.**

The value of $$x=5$$.

**Example 2:** Find the value of $$x$$ and $$y$$ from the equations $$x+y=8$$ and $$x-y=4$$.

**Step 1: Subtract $$y$$ from the first equation.**

$$x=8-y$$.

**Step 2: Put the obtained value of $$x$$ in the second equation.**

$$8-2y=4$$.

**Step 3: Multiply both the sides by $$-1$$.**

$$2y-8=-4$$

**Step 4: Add $$8$$ to both the sides to the above equation.**

$$2y=4$$.

**Step 5: Divide both the sides of the above equation by $$2$$.**

$$y=2$$.

**Step 6: Put the value of $$y$$ in the first equation.**

$$x=6$$.

**Step 7: Write the final answer.**

The value of $$x=6$$ and $$y=2$$.