The Distributive Property
A Linear equation is an equation in which the highest exponent of the variable present in the equation is one. When we draw the graph of the linear equation, it forms a straight line.
Example: x + 3 = 7, which is linear equation since the highest power of the variable, x present in the equation is 1.
How many solutions are there to a linear equation?
The solution of a linear equation is a value of the variable which satisfies the linear equation i.e. which makes the linear equation true.
A linear equation can have one, zero or infinite solution. We illustrate this with some examples.
- Linear equations with one solution.
Example 1: Consider the equation 7x – 35 = 0.
On solving we have, 7x = 35 or x = 5. The above linear equation is only true if x = 5.
Hence the given linear equation has only one solution i.e. x = 5.
- Linear equations with zero solution i.e. no solution.
Example 2: Consider the equation 7x – 35 = 5x + 2x – 27.
On solving we have, 7x – 35 = 7x – 27
Subtracting 7x from both sides, 7x – 7x – 35 = 7x – 7x – 27
We have -35 = -27, which is a false statement since it can’t be true for any value of the variable x.
Hence, the given linear equation has zero solution or the number of solutions is zero.
- Linear equations with infinite solutions.
Example 3: Consider the equation 25x – 35 = 5(5x + 4) – 55.
On solving we have, 25x – 35 = 25 x + 20 – 55 or 25x – 35 = 25x – 35.
Subtracting 25x from both sides, 25x – 25x – 35 = 25x – 25x – 35
We have -35 = -35, which is a true statement and it will be true for any value of the variable x. Hence the given linear equation has Infinite solutions or the number of solutions is infinite.
There are different forms of writing a linear equation. There is a way for solving the different forms of linear equations. For solving equations, we try to separate the variable to find the value of the variable. This is done using inverse operations.
if we add on left side of the equation,then same is added on the right side.A similar approach is followed,if we subtract,multiply or divide on one side, the same is applied on the other side.
Example: x + 3= 7
Subtract 3 from both sides , x + 3 – 3 – 7 – 3
So, x = 4
Here x=4 is the value of the unknown variable x, Which makes the equation true as
4 + 3 = 7.
When should you use the distributive property when solving an equation?
Distributive property is used to solve linear equations before separating the variables and it makes the process of finding the solution easier.
1) a(b + c) = ab + ac
2) a(b – c) = ab – ac
Example 1: Solve for x: 3(x – 5) = 24
Using the distributive property, 3(x) – 3(5) = 24
3x – 15 = 24
Adding 15 on both sides, we have 3x – 15 + 15 = 24 + 15
3x = 39
Dividing both sides by 3, x = 13
Example 2: Solve for x: 2(x + 7) + x = 26
Using the distributive property, 2(x) + 2(7) + x = 26
2x + x + 14 = 26 or 3x + 14 = 26
Adding 14 on both sides, 3x + 14 – 14 = 26 – 14
3x = 12
Dividing both sides by 3,x = = 4 .
Example 3: A gym charges a $50 activation fee and $17 per month for a membership. If twice the total money you spend is $ 474, for how many months, do you have gym membership?
Let x be the number of months for which the membership of gym is taken.
Then we have, 2(17x + 50) = 474
Using the distributive property, 2(17x) = 2(50) = 474
34x + 100 – 100 = 474 – 100
Subtract 100 on both sides 34x = 374
Divide by 34 on both sides, x = = 11
Hence the gym membership was taken for 11 months.
Solve each of the equation using distributive property and combining like terms.
- 2(x + 7) + 5x = 28
- 2(y – 1) + 3y = 3
- 2(3z – 1) + 2(4z + 5) = 8
- 8 + 3(x + 2) = 5(x – 1)
- A gift pack contains 3 identical t-shirts and a hat. If the hat costs $21 and Martin buys two such gift packs for $120. Find the cost of each t-shirt.
- A company sells their laptops at a price of $50 more from the manufacturing cost. If they sell 20 laptops for a total cost of $3,000, find the manufacturing cost of each laptop.
- A TV sports channel subscription costs $10 per month plus a $25 sign-up fee. If Martin paid a bill of $75, how many months subscription did he purchase?
- x = 2
- y = 1
- z = 0
- x = 19/2
- Cost of each t-shirt = $13
- Manufacturing cost of each laptop = $100.
- Martin purchased 5 months subscription.
Find out how eTutorWorld can help you improve your Grades.
No credit card is required, nor are you under any obligation to make a purchase. Just schedule the FREE TRIAL lesson to meet a tutor & get help on any topic you want!
Our Learning by Design methodology focuses exclusively on individual students.
Our expert tutors are specially trained to identify and diagnose the needs and skills of each student and plan future tutoring lessons accordingly.
Know more about our Personalised Online Tutoring Packs.
Give Your Child The eTutorWorld Advantage
Research has proven that personal online tutoring not just cements school learning, it helps build student confidence. eTutorWorld provides the best K-12 Online Tutoring Services so you can learn from the comfort and safety of your home at an affordable cost.
Be it an exam, class test or a quiz, eTutorWorld’s Math, Science and English tutors are responsible for your academic progress. Meet your personal coach at your convenient day and time to get help for Grade 3-12 Math, Science and English subjects and AP, SAT, SSAT and SCAT Test Prep help and test practice. All our tutors are graduates and bring with them years of teaching experience to the tutoring lessons.
Our ‘Learning by Design’ methodology makes sure that each student is at the center of the teaching-learning process. All tutoring sessions start with a question to the student ‘What do you want to learn today?’ Hence, tutors diagnose your skills and recognize your requirements before the actual tutoring happens. Post every tutoring session, an individualized worksheet is emailed to the student to assimilate learned concepts. Regular formative assessments are used to evaluate a student’s understanding of the subject.
The state of the art technology used is stable, user friendly and safe. All you need is a computer or a tablet and an internet connection. The easy-to-use web conferencing software requires a one time download, using which the student can talk and chat to the tutor, annotate on an interactive shared whiteboard or even share documents, assignments or worksheets.
All tutoring sessions are recorded and made available for a month so you can review concepts taught.
Email or call our support team with any issues or questions – we are here for you 24X7.
Also, download free printable math and science worksheets in pdf format and solve SCAT and SSAT Practice Tests online. Sign up for a Free Trial Lesson Today!
Thousands have taken the eTutorWorld Advantage – what are you waiting for?
© 2019 eTutorWorld - Online Tutoring and Test Prep | All rights reserved