The solutions here are so long. We are subtracting, adding, dividing, and multiplying both side by certain values. But instead simply we can take positive values to the other side and change their signs to make it easy. Is it ok in SAT as I have studied like it for a long time.

they don't ask for your work on the SAT, as long as your answer is correct they don't care how you get to it.

I finished foundations and am moving on to medium. How come it says that I have completed all the videos and problems? I thought there would be more content in this course.

So the thing is, the videos and articles are the same for each level. (Foundations, medium, advanced) Only the questions change. They get harder and more complex. Does this help?

Are the practice problems in these Khan Academy modules adaptive, the way the problems on the actual digital SAT are? In other words, if I get a 1/4 on a batch of problems will the next batch of the same skill be easier than if I got a 4/4?

nah, you'll get the "medium" problems every time you do a "medium" practice, as with the "easy" and "difficult" practices

Hello, I lack some of the math lessons on the sat ( there are lessons for math that I haven't even learned yet ). Does the articles and videos in the digital sat math here in khan academy covers everything for every lesson (even the basics) ?

Sadly No, But you can Manually search each of the topics

How about 2 solution and how do we figure it out?

Linear equations can't have 2 solutions

can you please explain the terms 'fraction coefficient' and 'fraction constant'

A coefficient is a number that is multiplying the variable (EX: *3*x +7=2) and if that coefficient is a fraction, (EX *1/2*x +7=2) it is a fraction coefficient. A constant is a number that is added or subtracted that is not multiplied or divided by the variable. A 'fraction constant' is a constant that is a fraction.

Main content

Course: Digital SAT Math > Unit 6

Lesson 1: Solving linear equations and inequalities: medium

Solving linear equations and linear inequalities | Lesson

Google Classroom

A guide to solving linear equations and linear inequalities on the digital SAT

What are linear equations and inequalities?

Linear equations and inequalities are composed of

and

Linear equations use the equal sign (

=

Linear inequalities use inequality signs (

>

<

\geq

, and

\leq

In this lesson, we'll learn to:

Solve linear equations
Solve linear inequalities
Recognize the conditions under which a linear equation has one solution, no solution, and infinitely many solutions

Note: If you're taking the SAT, then chances are you have a good understanding of how to solve linear equations and inequalities. However, it's important to solve them in their various forms with consistency. We recommend that you write out your steps (instead of doing everything in your head) to avoid careless errors, and we will do the same in our examples!

You can learn anything. Let's do this!

How do I solve linear equations?

Reasoning with linear equations

Khan Academy video wrapper

Reasoning with linear equationsSee video transcript

Types of linear equations

The goal of solving a linear equation is to find the value of a variable; we isolate the variable step by step until only the variable is on one side of the equation and only a constant is on the other.

When solving linear equations, the most important thing to remember is that the equation will remain equivalent to the original equation only if we always treat both sides equally: whenever we do something to one side, we must do the exact same thing to the other side.

Linear equations in one variable

Most of these questions on the SAT contain only one variable.

Example: If

2 x + 1 = 5

, what is the value of

x

We may be asked to combine like terms and distribute coefficients when solving.

When combining like terms, recall that:

a x \pm b x = (a \pm b) x

Example: If

2 x - 4 = 5 - x

, what is the value of

x

When distributing coefficients, recall that:

a (b x \pm c) = a b x \pm a c

Example: If

2 (x + 1) = 5

, what is the value of

x

Fractions and negative numbers

The presence of fractions and negative numbers can make linear equations more difficult to solve.

When solving a linear equation with fraction coefficients or constants:

If the equation has only a fraction coefficient, consider leaving the fraction until the last step in isolating $x$ ‍.

Example: If

\frac{1}{2} x + 3 = 5

, what is the value of

x

If the equation has both fraction coefficients and fraction constants, consider getting rid of the fractions in the first step.

Example: What is the solution to the equation

\frac{1}{2} x + \frac{1}{3} = \frac{1}{5}

When working with negative numbers, remember that:

$negative \cdot negative = positive$ ‍
$positive \cdot negative = negative$ ‍

Example: If

- 2 (x - 5) = 1

, what is the value of

x

Linear equations in two variables

Sometimes, we're given an equation in two variables and we're told the value of one of the variables. Plug the value of the known variable into the equation and solve.

Example: If

2 x + 5 y = 1

and

x = 3

, what is the value of

y

Using linear equations to evaluate expressions

Sometimes, we'll be given a linear equation in one variable and be asked to evaluate a different expression containing the variable. We can approach this type of question in two ways:

Solve the linear equation, then plug the value of the variable into the expression to evaluate it.
Find the relationship between the equation and the expression, then evaluate the expression without solving for the variable.

Knowing the second approach is not required, though it may save you valuable time on test day.

Example: If

2 x + 1 = 5

, what is the value of

8 x + 4

Try it!

Try: identify the steps to solving a linear equation

7 - 3 x = 28

To solve the equation above, we can first

both sides of the equation to isolate the

x

-term.

Next, we can

both sides of the equation by

- 3

What is the value of

x

How do I solve linear inequalities?

Multi-step inequalities

Khan Academy video wrapper

Inequalities with variables on both sidesSee video transcript

Types of linear inequalities

The steps for solving linear inequalities are similar to those for solving linear equations. For inequalities, we have to pay attention to the direction of the inequality signs.

Linear inequalities that do not require reversing the inequality sign

When the coefficient of

x

is positive, the inequality sign maintains its direction when we divide by the coefficient to isolate

x

Example: What values of

x

satisfy the inequality

2 x + 1 > 5

Linear inequalities that require reversing the inequality sign

When the coefficient of

x

is negative, we must reverse the direction of the inequality sign when we divide by the coefficient to isolate

x

Example: What values of

x

satisfy the inequality

- 2 x + 1 > 5

\begin{aligned} - 2 x + 1 & > 5 \\ - 2 x + 1 - 1 & > 5 - 1 \\ - 2 x & > 4 \\ \frac{- 2 x}{- 2} & < \frac{- 4}{- 2} & reverse sign \\ x & < - 2 \end{aligned}

Notice that we switched the direction of the inequality sign when we divided both sides of the equation by

- 2

. We can verify the solution by picking a few values.

For

x = - 3

, which is less than

- 2

and satisfies the inequality:

\begin{aligned} - 2 (- 3) + 1 & \overset{?}{>} 5 \\ 7 & \overset{✓}{>} 5 \end{aligned}

For

x = 1

, which is greater than

- 2

and does not satisfy the inequality:

\begin{aligned} - 2 (1) + 1 & \overset{?}{>} 5 \\ - 1 & ≯ 5 \end{aligned}

Try it!

try: identify the steps to solving a linear inequality

- 2 x > 9 + 4 x

To solve the inequality above, we can first

both sides of the equation.

Next, we can

both sides of the equation by

- 6

and

x

- \frac{3}{2}

How do I alter the number of solutions for linear equations?

Note: Questions about the number of solutions for linear equations do not appear on every test.

Creating an equation with no solutions

Khan Academy video wrapper

Creating an equation with no solutionsSee video transcript

Creating an equation with infinitely many solutions

Khan Academy video wrapper

Creating an equation with infinitely many solutionsSee video transcript

How many solutions can a linear equation have?

Most linear equations on the SAT have exactly one solution. Linear equations with no solutions or infinitely many solutions must be engineered by specifying the values of constants.

For a linear equation in one variable:

If the equation can be rewritten in the form $x = a$ ‍, where $a$ ‍ is a constant, then that equation has one solution.
If the variable can be eliminated from the equation, and what remains is the equation $a = b$ ‍, where $a$ ‍ and $b$ ‍ are different constants, then the equation has no solution. (No value of $x$ ‍ can make $1$ ‍ equal to $2$ ‍!)
If the equation can be rewritten in the form $x = x$ ‍, then the equation has infinitely many solutions. (No matter what the value of $x$ ‍ is, it will always equal itself!)

Let's look at some examples!

2 x - 1 = a x - 2

a = 2

in the equation above, what value of

x

satisfies the equation?

2 x - 4 = a (x - 2)

a = 2

is the equation above, what value of

x

satisfies the equation?

Try it!

try: change the number of solutions for a linear equation

5 x + 3 = a x + b

In the equation above,

a

and

b

are constants.

The equation has a single solution if

a

and

b

is a real number.

The equation has infinitely many solutions if

a

and

b

Your turn!

Practice: evaluate a linear expression

6 x + 10 = 24

, what is the value of

3 x + 5

Practice: solve a linear equation in one variable

2 (3 x + 1) - (9 - 2 x) = 25

What value of

x

satisfies the equation above?

Practice: find a value that does not satisfy an inequality

Which of the following numbers is NOT a solution to the inequality

7 x - 3 \leq 3 x + 11

practice: determine the condition for no solution

3 a x - 11 = 4 (x + 2) + 2 (x - 1)

In the equation above,

a

is a constant. If no value of

x

satisfies the equation, what is the value of

a

Things to remember

When solving linear inequalities:

If the coefficient of $x$ ‍ is positive, the inequality sign maintains its direction when we divide by the coefficient to isolate $x$ ‍.
If the coefficient of $x$ ‍ is negative, we must reverse the direction of the inequality sign when we divide by the coefficient to isolate $x$ ‍.

When determining the number of solutions for a linear equation:

If the equation can be rewritten in the form $x = a$ ‍, where $a$ ‍ is a constant, then that equation has one solution.
If the variable can be eliminated from the equation, and what remains is the equation $a = b$ ‍, where $a$ ‍ and $b$ ‍ are different constants, then the equation has no solution.
If the equation can be rewritten in the form $x = x$ ‍, then the equation has infinitely many solutions. (No matter what the value of $x$ ‍ is, it will always equal itself!)

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muhammadhanzala544
Posted a year ago. Direct link to muhammadhanzala544's post “The solutions here are so...”
The solutions here are so long. We are subtracting, adding, dividing, and multiplying both side by certain values. But instead simply we can take positive values to the other side and change their signs to make it easy. Is it ok in SAT as I have studied like it for a long time.
Button navigates to signup pageComment on muhammadhanzala544's post “The solutions here are so...”
(61 votes)
Answer
- om.king6543
  Posted a year ago. Direct link to om.king6543's post “they don't ask for your w...”
  they don't ask for your work on the SAT, as long as your answer is correct they don't care how you get to it.
  Comment on om.king6543's post “they don't ask for your w...”
  (74 votes)
Test Taker
Posted 8 months ago. Direct link to Test Taker's post “*Hello*! Fellow test take...”
Hello! Fellow test takers, just like you all I am also preparing for my test, I wish you all ,and wish me too
Good luck out there test takers you've got this

Just completed this section It's easy if you have any doubts go ahead and drop'em ill try to answer to it as simply as possible
Button navigates to signup pageComment on Test Taker's post “*Hello*! Fellow test take...”
(62 votes)
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- serikbaysymbat35
  Posted 8 months ago. Direct link to serikbaysymbat35's post “thanks a lot, break a leg”
  thanks a lot, break a leg
  Comment on serikbaysymbat35's post “thanks a lot, break a leg”
  (14 votes)
talhasami928
Posted 9 months ago. Direct link to talhasami928's post “I finished foundations an...”
I finished foundations and am moving on to medium. How come it says that I have completed all the videos and problems? I thought there would be more content in this course.
Button navigates to signup pageButton navigates to signup page
(9 votes)
Answer
- StudyBuddy
  Posted 9 months ago. Direct link to StudyBuddy's post “So the thing is, the vide...”
  So the thing is, the videos and articles are the same for each level. (Foundations, medium, advanced) Only the questions change. They get harder and more complex.
  Does this help?
  Comment on StudyBuddy's post “So the thing is, the vide...”
  (29 votes)
prativakhanal1
Posted a year ago. Direct link to prativakhanal1's post “gotta start again for aug...”
gotta start again for aug sat
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(15 votes)
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- Rayana
  Posted 10 months ago. Direct link to Rayana's post “Same here.”
  Same here.
  Comment on Rayana's post “Same here.”
  (7 votes)
Alanood Alm
Posted a year ago. Direct link to Alanood Alm's post “when solving for x just u...”
when solving for x just use the calculator since it is allowed on the whole math section you can just type the equation
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- 108026
  Posted 5 months ago. Direct link to 108026's post “Thanks bud”
  Thanks bud
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fardinn78
Posted a year ago. Direct link to fardinn78's post “How about 2 solution and ...”
How about 2 solution and how do we figure it out?
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(4 votes)
Answer
- Mrsosh
  Posted a year ago. Direct link to Mrsosh's post “Linear equations can't ha...”
  Linear equations can't have 2 solutions
  Comment on Mrsosh's post “Linear equations can't ha...”
  (21 votes)
arshiashaikh
Posted a year ago. Direct link to arshiashaikh's post “can someone who gave the ...”
can someone who gave the digital test please share their experience
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(13 votes)
Answer
- jasmine sara
  Posted 3 days ago. Direct link to jasmine sara's post “It was generally okay. It...”
  It was generally okay. It's just like the bluebook practice test. But the time in the second module goes really fast if you get the harder one, so you need to practice and master time saving tips
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  (1 vote)
noormunnazah
Posted 8 months ago. Direct link to noormunnazah's post “can you please explain th...”
can you please explain the terms 'fraction coefficient' and 'fraction constant'
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(2 votes)
Answer
- S B
  Posted 8 months ago. Direct link to S B's post “A coefficient is a number...”
  A coefficient is a number that is multiplying the variable (EX: *3*x +7=2) and if that coefficient is a fraction, (EX *1/2*x +7=2) it is a fraction coefficient. A constant is a number that is added or subtracted that is not multiplied or divided by the variable. A 'fraction constant' is a constant that is a fraction.
  Button navigates to signup page
  (16 votes)
gspeidel
Posted 10 months ago. Direct link to gspeidel's post “Are the practice problems...”
Are the practice problems in these Khan Academy modules adaptive, the way the problems on the actual digital SAT are? In other words, if I get a 1/4 on a batch of problems will the next batch of the same skill be easier than if I got a 4/4?
Button navigates to signup pageButton navigates to signup page
(5 votes)
Answer
- Aubrey
  Posted 9 months ago. Direct link to Aubrey's post “nah, you'll get the "medi...”
  nah, you'll get the "medium" problems every time you do a "medium" practice, as with the "easy" and "difficult" practices
  Button navigates to signup page
  (3 votes)
tydinis69
Posted 10 months ago. Direct link to tydinis69's post “Hello, I lack some of the...”
Hello, I lack some of the math lessons on the sat ( there are lessons for math that I haven't even learned yet ). Does the articles and videos in the digital sat math here in khan academy covers everything for every lesson (even the basics) ?
Button navigates to signup pageComment on tydinis69's post “Hello, I lack some of the...”
(5 votes)
Answer
- Aravind
  Posted 10 months ago. Direct link to Aravind's post “Sadly No, But you can Man...”
  Sadly No, But you can Manually search each of the topics
  Comment on Aravind's post “Sadly No, But you can Man...”
  (2 votes)