Solve the following equations with decimals. 4.1 x + 2.4 = 3.6 x − ( − 4 ) 4.1x+2.4=3.6x-\left(-4\right)

Welcome back. In this problem, we wanna solve the equation 4.1 x plus 2.4 equals 3.6 x minus negative four. Now this equation has decimals, so we have to start with step zero, clearing those decimals by multiplying by a power of 10. We can check each of our terms and see how many decimal places it has. Then when we find the highest number, we'll multiply by that power of 10. Let's start on the left side of the equation. We have 4.1 x, and 4.1 has one decimal place. Next, we have 2.4, which also has one decimal place. On the right side of the equation, we have 3.6 x, and 3.6 has one decimal place. And finally, negative four has zero decimal places. So looking at our number of decimal places, we have one, one, one, and zero, so the highest number there is one, and we want to multiply by the first power of ten, ten to the first, which is just 10. So let's multiply both sides of our equation by 10. We get 10 times 4.1 x plus 2.4 is equal to 10 times 3.6 x minus negative four. Let's distribute that 10 on both sides of the equation. On the left side, we get 10 times 4.1 x first. Now when we multiply by 10, we move the decimal point one place to the right. So when we do 10 times 4.1, we move that decimal point one place to the right to get 41, and that becomes 41 x. Next, we multiply 10 by 2.4, and again, we move the decimal point one place to the right to get 24. On the right side of the equation, we'll do 10 times 3.6 x, and we move the decimal point one place to the right. 10 times 3.6 is 36, so that gets us 36 x. Next, we do 10 times negative four, which is minus 40, so we get minus negative 40. Now that we have no more decimals, step zero is all set and we move on to step one. This is a great time to simplify our minus negative 40. As a reminder, minus a negative is just a positive, so we can rewrite this as 41 x plus 24 is equal to 36 x plus 40. We don't have any more distributing or combining like terms we have to do, so step one is all set, and we can move on to step two. We wanna collect all of our variable terms on one side of the equation and all of our constant terms on the other side. Currently, we have variable and constant terms on both sides of the equation. So let's try to get variables to the left and constants to the right, starting with our constants. We need to get rid of this plus 24 on the left, so we can do that by subtracting 24 from both sides of the equation. It cancels on the left, leaving us with 41 x, and on the right side, we get 36 x plus 40 minus 24, which is plus 16. Next, we wanna get rid of the 36 x on the right so that all of our variable terms are on the left side of the equation. So we want to subtract 36 x from both sides of the equation. That's going to cancel on the right side of the equation, leaving us with 16. On the left side of the equation, we have 41 x minus 36 x, which is five x. Now that we have all of our variable terms on the left side of the equation and constants on the right, step two is all set, and we move on to step three where we want to isolate our variable. We need to get rid of this coefficient of five so we can divide both sides of the equation by five. It cancels the coefficient on the left side leaving us with x. On the right side, we can't simplify the fraction 16 fifths, so we get the solution x equals 16 fifths. When our equation has decimals, sometimes we like to express the answer as a decimal instead. We can get the decimal by doing 16 and dividing it by five, and if you do, you'll get x equals 3.2. Three point two and sixteen fifths are equal to each other, so both of these solutions are the same solution, just expressed in a different way. We can see that our variable is isolated, so step three is all set. For step four, you can choose one of these forms, 16 fifths or 3.2, and plug that value of x back into the original equation to verify our solution. But our solution is 16 fifths, which is equal to 3.2. Once you're all set checking, great job.