Question

Solve the given linear equation \[16\left( {3x - 5} \right) - 10\left( {4x - 8} \right) = 40\]

Answer 1

Hint: Here the given equation is a linear equation. A linear equation is an equation in which the highest degree is one. First, we will multiply the terms on the left hand side of the equation using the distributive property. Then we will take the terms with \[x\] on one side of the equation and constants to the other side of the equation. We will then solve the equation to get the value of \[x\].

Complete step-by-step answer:
The given equation is \[16\left( {3x - 5} \right) - 10\left( {4x - 8} \right) = 40\].
First, we will multiply the numbers with the terms inside the brackets using the distributive property of multiplication. Therefore, we get
 \[ \Rightarrow 16\left( {3x} \right) - 16\left( 5 \right) - 10\left( {4x} \right) - 10\left( { - 8} \right) = 40\]
\[ \Rightarrow 48x - 80 - 40x + 80 = 40\]
Now we will keep all the terms with \[x\] on the LHS of the equation and all the constants on the RHS of the equation. Therefore, we get
\[ \Rightarrow 48x - 40x = 40 + 80 - 80\]
Now we will solve this equation to get the value of \[x\].
Adding and subtracting the like terms, we get
\[ \Rightarrow 8x = 40\]
Dividing both sides by 8, we get
\[ \Rightarrow x = \dfrac{{40}}{8}\]
\[ \Rightarrow x = 5\]
Hence, the value of \[x\] is 5.

Note: Here, we have used distributive property which states that if there is an expression of the form $a\left( {b + c} \right)$ then it is equal to $ab + bc$. That is by the distributive property of multiplication $a\left( {b + c} \right) = ab + bc$. Also we have got only one solution of $x$ because a linear equation has only one solution. This is because for any equation numbers of roots are always equal to the value of the highest exponent of the variable.
In the similar manner, the quadratic equation has 2 solutions because its highest degree is 2. Also, the cubic equations are the equation in which the highest exponent of the variable is 3 and so it has 3 solutions.
We can represent quadratic and cubic equation in the following form:
Quadratic equation: \[a{x^2} + bx + c = 0\]
Cubic equation: \[a{x^3} + b{x^2} + cx + d = 0\]