Question

Question: Show that $\mathbf{6}\mathbf{x}\mathbf{+}\mathbf{5}\equiv \mathbf{7}\mathbf{\left(}\mathbf{mod}\mathbf{5}\mathbf{\right)}$ has infinitely more solutions [Hint: - Exercise 1 and Exercise 2 ].

Short answer

Answer:

Any v which satisfies the property that $\text{v}\equiv 2\text{(mod 5)}$ will be a solution. Hence, $\text{6x + 5}\equiv \text{7(mod 5)}$ has infinitely more solutions.

Step-by-step solution

Concept Introduction

Both sides of an infinite solution are equal.As an illustration, take equation$6x+2y-8=12x+4y-16$. If a formula or method for infinite solutions is used to simplify the equation, it will be obtained that both the sides are equal, making it an infinite solution. The concept of infinite denotes boundlessness or infinity.

Determine RHS value of Equation

Recall that if$\text{u = v}\left(\text{mod n}\right)$and ifis a solution of the given equation, thenis also its solution.

The equation is$\text{6x + 5}\equiv \text{7(mod 5)}$, the value$\text{7(mod 5)}$ at the right-hand side of the equation is . The value$\text{7(mod 5)}$is 2 , hence the RHS of the equation is 2.

$$\begin{gathered} \text{6x + 5}\equiv \text{7(mod 5)} \\ \text{6x + 5}\equiv 2 \end{gathered}$$

Finding LHS using Substitution method

Choose the value x to make the RHS equal to the LHS,

Substitute$x=-\frac{1}{2}$

$$\begin{gathered} 6\left(-\frac{1}{2}\right)+5=2 \\ -3+5=2 \\ 2=2 \end{gathered}$$

is a solution of the equation$6x+5\equiv 7\left(\mathrm{mod}5\right)$.

Therefore, any v which satisfies the property that $v\equiv 2\left(\mathrm{mod}5\right)$ will be a solution.