Video Transcript
What value of 𝑥 satisfies the equation negative four multiplied by three 𝑥 minus 13 plus eight 𝑥 minus 12 equals negative 20?
Our first step is to distribute the parentheses, otherwise known as expanding the bracket. We multiply negative four by three 𝑥 and negative four by negative 13. Negative four multiplied by three 𝑥 is equal to negative 12𝑥. And negative four multiplied by negative 13 is equal to positive 52. Multiplying two negative numbers together gives us a positive answer.
As there is no number outside the second parentheses, we can drop down the eight 𝑥 and negative 12. We can also bring down the negative 20 on the other side of our equal sign. Our next step is to group, or collect, the like terms. We can group the 𝑥 terms and the constants on the left-hand side. Negative 12𝑥 plus eight 𝑥 is equal to negative four 𝑥. And 52 minus 12 is equal to 40. The equation simplifies to negative four 𝑥 plus 40 is equal to negative 20.
We need to calculate the value of 𝑥. We can, therefore, subtract 40 from both sides of the equation. On the left-hand side, the 40s cancel. And on the right-hand side, negative 20 minus 40 is equal to negative 60. Finally, we need to divide both sides of the equation by negative four. Negative four divided by negative four is equal to one. So, on the left-hand side, we are left with 𝑥. On the right-hand side, negative 60 divided by negative four is equal to 15. As with multiplying, when we divide two negative numbers, we get a positive answer. The value of 𝑥 that satisfies the equation is 15.
We can check this answer by substituting our value of 𝑥 back into the original equation. We substitute 15 everywhere that 𝑥 appears in the question. This gives us negative four multiplied by three multiplied by 15 minus 13 plus eight multiplied by 15 minus 12. Three multiplied by 15 is equal to 45. And eight multiplied by 15 is equal to 120. 45 minus 13 is equal to 32. And 120 minus 12 is equal to 108. We are left with negative four multiplied by 32 plus 108. Negative four multiplied by 32 is equal to negative 128. Adding 108 to this does indeed give us negative 20. Therefore, our answer of 𝑥 equals 15 is correct.
There is a quicker method, we might have noticed from the initial question, as all three terms are divisible by four or negative four. Let’s see what happens when we divide each of the three terms by negative four. In the first term, the negative fours cancel. So, we are left with three 𝑥 minus 13. Both eight 𝑥 and negative 12 are divisible by negative four. Eight 𝑥 divided by negative four is equal to negative two 𝑥. And negative 12 divided by negative four is equal to three.
On the right-hand side, negative 20 divided by negative four is equal to five. We can once again collect the like terms, the 𝑥s and the constants. Three 𝑥 minus two 𝑥 is equal to one 𝑥, or just 𝑥. And negative 13 plus three is equal to negative 10. This leaves us with 𝑥 minus 10 is equal to five. We can then add 10 to both sides of this equation. Once again, this gives us a final answer of 𝑥 equals 15.
