Question

Solve the following equations:
I. \[5x - 12 = 2x - 6\]
II. \[7p - 3 = 3p + 8\]

Answer 1

Hint: Simultaneous linear equations are two equations, each with some unknowns and are simultaneous because they are solved together. The algebraic expression should be any one of the forms such as addition, subtraction, multiplication and division, hence the given equations are linear as there are constant variables involved and to solve the given inequality, combine all the like terms and then simplify the terms to get the value of \[x\] and p.

Complete step by step solution:
Let us solve the first equation:
I. \[5x - 12 = 2x - 6\]
To get the combined terms, add 12 on both the sides of the given equation
 \[5x - 12 + 12 = 2x - 6 + 12\]
As we can see that -12 and +12 implies zero.
 \[5x = 2x - 6 + 12\]
Simplifying the numbers in the equation, we get
 \[5x = 2x + 6\]
Subtract 2x from both sides of the equation
 \[5x - 2x = 2x + 6 - 2x\]
As the equation consists of like terms, so let us combine all the like terms and simplify it
 \[3x = 2x + 6 - 2x\]
 \[ \Rightarrow \] \[3x = 6\]
Now divide both sides of the equation by the same term i.e., 3 we get
 \[\dfrac{{3x}}{3} = \dfrac{6}{3}\]
We can see that the numerator term and denominator term both are the same which implies one with the remaining \[x\] term, as we need to get the value of \[x\] . Hence, we get
 \[x = \dfrac{6}{3}\]
Therefore, we get the value of \[x\] as
 \[x = 2\]
Hence, the value of \[x\] in the given equation \[5x - 12 = 2x - 6\] is \[x = 2\] .
So, the correct answer is “ \[x = 2\]”.

Now let us solve the second equation:
II. \[7p - 3 = 3p + 8\]
To get the combined terms, add 3 on both the sides of the given equation
 \[7p - 3 + 3 = 3p + 8 + 3\]
As we can see that -3 and +3 implies zero.
 \[7p = 3p + 8 + 3\]
Simplifying the numbers in the equation, we get
 \[7p = 3p + 11\]
Subtract 3p from both sides of the equation
 \[7p - 3p = 3p + 11 - 3p\]
As the equation consists of like terms, so let us combine all the like terms and simplify it
 \[4p = 3p + 11 - 3p\]
 \[ \Rightarrow \] \[4p = 11\]
Now divide both sides of the equation by the same term i.e., 4 we get
 \[\dfrac{{4p}}{4} = \dfrac{{11}}{4}\]
We can see that the numerator term and denominator term both are the same which implies one with the remaining p term, as we need to get the value of p. Hence, we get the value of p as
 \[p = \dfrac{{11}}{4}\]
Hence, the value of p in the given equation \[7p - 3 = 3p + 8\] is \[p = \dfrac{{11}}{4}\] .
So, the correct answer is “ \[p = \dfrac{{11}}{4}\]”.

Note: The key point to solve this type of equation is to combine all the like terms i.e., finding out the common term and evaluate for the variable asked. Equations that have more than one unknown can have an infinite number of solutions, finding the values of letters within two or more equations are called simultaneous equations because the equations are solved at the same time.