Linear equations are expressions that can reduce to the form ax+b=0, with a≠0. Their only solution is x=-(b/a) and the unknown x only appears raised to 1.
Equivalent equations
Two equations are equivalent if they have the same solution or both have no solution. Thus, the equations 5x–9=51 and 3x-7=89–5x are equivalent because the solution of both is x=12.
To solve an equation, we have to clear the x using a series of steps. Each step consists of transforming the equation into another equivalent equation, in which the x is closest to being cleared.
How to solve linear equations
Keep in mind the following rules to avoid making mistakes:
- Add or subtract the same expression in the two members of the equality. That is, what is adding in one member passes by subtracting the other member, and vice versa.
- Multiply or divide the two members by the same non-zero number. In other words, what is multiplying everything else from one member happens dividing the other, and vice versa.
How to solve a linear equation:
- Remove denominators, if any.
For this, the two members of the equation are multiplied by a common multiple of the denominators; preferably, their least common multiple:
- Remove parentheses, if any.
- Pass the x terms to one member and the numbers to the other member:
- Simplify each member:
- Clear the x and get the solution like this:
- Check by substituting the x for the solution obtained.
Since they match, the solution is correct:
The best way to learn to solve linear equations is to practice doing lots of exercises. Start now and test yourself!
Linear equation exercises
1. Find out the value of x:
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1. Question
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a) 2x - 5 + 3x + 1 = 3x - 2 → x =
b) x + 7 = 12x - 3 - 8x + 1 → x =
c) 6x - 1 + x = 4 + 3x + 3 → x =
d) x + 2x + 3x - 5 = 4x - 9 → x =
e) 2x - 8x + 3x + 2 = 4x - 6x + 5 → x =
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2. Solve the following equations:
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1. Question
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a) 2(2 – 3x) – 3(3 – 2x) = 4(x + 1) + 3(4 – 5x) → x =
b)x - 3 5=x - 1 3- 2 → x =
c)x + 3 3-x 2= 1 → x =
d)3x + 4 5=x + 2 2→ x =
e)5x - 16 6= -x+8 12+x+1 3→ x =
f)2x - 4 3= 3 -4 - x 2→ x =
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a) 2(2 – 3x) – 3(3 – 2x) = 4(x + 1) + 3(4 – 5x) → x =
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