8 th Grade Unit 5 Lesson 3 Equations for Functions Illustrative Mathematics

eighth grade unit five lesson three equations for functions problem number one here is an equation that represents function seventy-two plus twelve equals sixty select all the different equations that describe the same function let's look at the term on the right side of the equal sign it's 600 and compared to the original equation that term used to be 60 so it's actually 10 times bigger than it used to be look at the term it went from 12 times 10 and became 120 and now let's look at the term it went from 72x to 720 it was multiplied by 10 so all three terms were multiplied by 10. another important thing that noticed is all the terms that were on the left side of the equal sign remained on the left side of the equal sign and all the terms that were on the right side of the equal sign remained on the right side of the equal sign this next equation will tell us why that's important let's look at this term to the left of the equal sign there's one but in the original equation there used to be 12y 12y divided by 12 equals 1y now let's look at the term with the it's moved from one side of the equal sign to the other side of the equal sign so not only did the 72x get 12 times smaller by being divided by 12 but they subtracted it from both sides of the equal sign so 72x divided by 12 equals 6x but if we move it to the other side of the equal sign it's because we subtracted that 6x from both sides so if we're subtracting 6x on the right hand side we'd have negative 6x and it would be cancelled out from the left hand side the only term left is 60 and it turned into 5 because 60 divided by 12 equals 5 and 60 is still on the right hand side of the equal sign and the 5 is on the right hand side of the equal sign so this equation follows the same function as the original equation let's look at the term on the right side of the equal sign it went from 60 to ten that means it was divided by six or got six times smaller sixty divided by six is ten now let's go to the other side of the equal sign we see twelve twelve divided by six is two and then finally also on the left hand side of the equal sign is 72x let's divide that by 6 and we get 12x the equation for also shares the same function as the original equation equals 5 plus 6 let's look at the term on the left hand side of the equal sign it started out as 12 and then became 1y so that's 12 times smaller so 12y divided by 12 equals 1y now let's look at the term the term actually went from the left hand side of the equation to the right hand side of the equation 72x divided by 12 equals 6x however if we were to subtract 6 from both sides it would end up as negative 6x on the right hand side of the equal sign says it's positive 6 but it should be negative 6 so does not follow the same function as the original equation let's look at the term in the original equation it started out as 72x and then became 1x so it was obviously divided by 72. let's look at the 60 in the original equation it started as 60 then was divided by 72 and became 5 6 and 60 over 72 is equivalent to 5 over 6. so there's one term left that's the term with the in it it starts out on the left side of the equal sign and then ends up on the right side of the equal sign so let's see if 12y divided by 72 then subtracted from both sides of the equal sign is indeed negative over six let's find out we can subtract 12y divided by 72 from both sides it gets cancelled out from the left side and this is what's left on the right side negative 12y divided by 72 is the same as negative 12 over 72 12 goes into negative 12 evenly and 12 goes into 72 evenly so we can divide the top and the bottom by 12. negative 12 divided by 12 is and 72 divided by 12 is 6. so does follow the same function as the original equation let's look at the term on the right side of the original equation sixty sixty becomes six sixty divided by ten equals six so let's see if the other terms are also ten times smaller seventy-two divided by ten does not equal 7x and 12y divided by 10 does not equal 2y so the equation for does not follow the same function as the original equation equals five over six plus over six seventy-two in the original equation divided by seventy-two equals one let's divide sixty by seventy-two 60 over 72 represents 60 divided by 2. 12 goes into 60 evenly and 12 goes into 72 evenly 60 divided by 12 is 5 and 72 divided by 12 is 6. 60 divided by 72 is equivalent to 5 over 6. let's look at the term with in it 12y it was subtracted from both sides so it should be negative on the right hand side of the equal sign but when we look at it's written as positive equation does not represent the function of the original equation problem number two from eighth grade unit four lesson 13 graph system of linear equations with no solutions this means need to graph two lines that have the same slope but different y-intercepts the y-intercept for this line is 3 and the y-intercept for the next line is 6. they have the same slope and different y-intercepts write an equation for each line you graph can start out the equations for both of these lines with equals next we need to figure out what the slope is the rise is up one and the run is to the right one slope of one over one is the same as slope of one for both equations this middle term will be one over one times or one times now we need to find the last term for each of these equations this is where the equations will be different for the blue line the y-intercept is 3 so the equation reads equals 1m plus 3. for the purple line the y-intercept is six so the equation reads equals one plus six you can rewrite these equations as equals plus three and equals plus six problem number three brown rice costs two dollars per pound and beans cost one dollar and sixty cents per pound lynn has ten dollars to spend on these items to make large meal of beans and rice for potluck dinner let be the number of pounds of beans lynn buys and be the number of pounds of rice she buys when she spends all her money on this meal write an equation relating the two variables rice at two dollars per pound can be expressed as two and beans at dollar sixty per pound can be expressed as one point six so the equation would read two plus 1.6 equals 10. rearrange the equation so is the independent variable to get the by itself we have to divide every term by two two divided by two is one ten divided by two is five now we can divide the independent variable by two one point six divided by two equals zero point eight since we need the independent variable on the other side of the equal sign we have to subtract 0.8 from both sides now the equation reads equals 5 minus 0.8 rearrange the equation so is the independent variable let's start with the original equation 2r plus 1.6 equals 10. now we want as the independent variable let's get the on the other side of the equal sign by subtracting 2r from both sides 2r minus 2r cancels each other out next let's get the by itself so that it's just 1b and we'll do that by dividing both sides by 1.6 every term needs to be divided by 1.6 1.6 divided by 1.6 equals 10 divided by 1.6 equals 6.25 and negative 2r divided by 1.6 equals negative 1.25 that leaves us with the equation equals 6.25 minus 1.25 problem number four from eighth grade unit four lesson six solve each equation and check your answer use the distributive property to multiply four times three that's twelve and 4 times negative 2x that's negative 8x bring down the 2x and combine like terms 2x minus 8x is negative 6x now you have 12 minus 6x on the left side of the equal sign and on the right side of the equal sign use the distributive property to multiply 3 times 2 that's 6x and 3 times positive 2. that's positive 6 and that's over 6 so we have to divide both those terms by six six divided by six is one or and six divided by six is one combine like terms and one plus four is five let's add 6x to both sides negative 6x plus 6x cancels each other out and plus 6x equals 7x because that's like 1x plus 6x to get the 7x by itself let's subtract five from both sides five minus five cancels each other out and twelve minus five equals seven to make the just one we have to divide both sides by seven seven divided by seven is and 7 divided by 7 is 1. equals 1. to check your answer substitute 1 in place of all of the x's 4z plus 5 equals negative 3z minus 8. add 3z to both sides negative 3z plus 3z cancels each other out and 4z plus 3z equals 7z subtract 5 from both sides of the equal sign 5 minus 5 cancels each other out and negative 8 plus negative 5 equals negative 13. let's make the just one by dividing both sides by seven seven divided by seven equals one and negative thirteen divided by seven equals negative thirteen sevenths equals negative thirteen sevenths you can check your answer by substituting the with negative thirteen sevenths let's look at this term on the right minus one over four that's the same as over 4 minus 1 over 4. since we're subtracting fractions let's make common denominator of 8. so for the first term we need to multiply the numerator and the denominator by 4. so one-half becomes four eighths we need to multiply the numerator and the denominator times two so four becomes eight and becomes two cubed and for this last term on the right we need to multiply the numerator and denominator by two so the four becomes an eight and the one becomes two let's collect the q's just on one side of the equal sign and we can do this by adding one eighth to both sides negative 1 8 plus 1 8 cancels each other out and 2 over 8 plus 1 over 8 equals 3 over 8. to get the term with the in it by itself we have to add two eighths to both sides negative two eighths plus two eighths cancels each other out and four eighths plus two eighths equals six eighths to make the just one we have to multiply by the reciprocal of three-eighths so we have to multiply both sides by eight thirds three-eighths cubed times eight-thirds equals one and eight-thirds times six-eighths equals forty-eight twenty-fourths which is equal to two so equals two you can check your answer by substituting the with two help me disrupt youtube's algorithm by liking this video commenting on this video sharing this video and subscribing to my channel thanks appreciate it you