Video: Solving Linear Equations

What value of 𝑥 solves ((𝑥 + 1)/2) − ((𝑥 − 1)/3) = 𝑥?

02:44

Video Transcript

What value of 𝑥 solves 𝑥 plus one over two minus 𝑥 minus one over three is equal to 𝑥?

In order to solve this equation, we firstly need to consider the left-hand side. In order to add or subtract any fractions, we firstly need to make sure the denominators are the same. We do this by finding the lowest common multiple or LCM. In this case, this would be six. We need to multiply the numerator and denominator of the first fraction by three and the second fraction by two. The first fraction becomes three multiplied by 𝑥 plus one over six. The second fraction, which has been subtracted, becomes two multiplied by 𝑥 minus one over six. This is all equal to 𝑥. As the denominators are now the same, we can write the left-hand side as a single fraction.

Our next step is to distribute the parentheses, otherwise known as expanding the brackets. Three multiplied by 𝑥 is three 𝑥. And three multiplied by one is three. Negative two multiplied by 𝑥 is negative two 𝑥. And negative two multiplied by negative one is positive two. Remember, when we multiply a negative by a negative, our answer is positive. The equation simplifies to three 𝑥 plus three minus two 𝑥 plus two over six is equal to 𝑥. We can now simplify the numerator by collecting or grouping like terms. Three 𝑥 minus two 𝑥 is equal to 𝑥, and three plus two is equal to five. We have 𝑥 plus five over six is equal to 𝑥.

We can now solve this equation using the balancing method and our knowledge of inverse operations. We begin by multiplying both sides by six. This gives us 𝑥 plus five is equal to six 𝑥. As there is a higher coefficient of 𝑥 on the right-hand side, we can subtract 𝑥 from both sides. This gives us five is equal to five 𝑥. Finally, we divide both sides of this equation by five. The value that solves the equation is 𝑥 equals one. We could check this by substituting this value back in to the original equation.

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