Linear equations with one variable

An equation of the form ax = b, where x is a variable and a and b are some numbers, is called a linear equation with one variable.

The roots of linear equations with one variable.

If the coefficient a ≠ 0, then the solution to this equation is unique and equal to x = b/a.

If a = 0, and b ≠ 0, then the equation has no roots, since 0*x ≠ b for any b ≠ 0.

It remains to consider the case a = 0 and b = 0. In this case, any value of x will be a root of the equation, since the equality 0*x = 0 is true for any value x.

Thus, the linear equation ax = b has a unique root for a ≠ 0, has no roots for a = 0 and b ≠ 0, and has an infinite number of roots for a = 0 and b = 0.

Examples of solving linear equations with one variable.

Example 1.
Solve the equation x = -x.
Solution.
x = -x   <=>   x + x=0   <=>   2x=0   <=>   x=0/2   <=>   x=0.
Answer: x = 0.

Example 2.
Solve the equation 3(2.5 - 2x) = 5 - 6x.
Solution.
3(2.5 - 2x) = 5 - 6x   <=>   7.5 - 6x = 5 - 6x   <=>   -6x + 6x = 5 - 7.5   <=>   0*x = -2.5.
But 0 ≠ -2.5 for any x. Therefore, the equation has no roots.
Answer: there are no roots.

Example 3.
Solve the equation 8x = 6 + 2(4x - 3).
Solution.
8x = 6 + 2(4x - 3)x   <=>   8x = 6 + 8x - 6   <=>   8x - 8x = 6 - 6   <=>   0*x = 0. We have obtained an identity that is true for any values ​​of x. Therefore, the root of the equation is any number.
Answer: x - any number.

Example 4.
Solve the equation 3x - (10 + 5x) = 54.
Solution.
3x - (10 + 5x) = 54   <=>   3x - 10 - 5x = 54   <=>   -2x = 54 + 10   <=>   -2x = 64   <=>   x = -32.
Answer: x = -32.

Example 5.
Solve the equation 6x - (x - 1) = 4 + 5x.
Solution.
6x - (x - 1) = 4 + 5x   <=>   6x - x + 1 = 4 + 5x   <=>   5x - 5x = 4 - 1   <=>   0 = 3.
But 0 ≠ 3 for any x. Therefore the original equation has no roots.
Answer: there are no roots.

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