##What are linear equations and inequalities?

Linear equations and inequalities are composed of [[☃ definition 1]] and [[☃ definition 2]].

Linear equations use the equal sign ($=$). [[☃ explanation 1]]

Linear inequalities use inequality signs ($>$, $<$, $\geq$, and $\leq$). [[☃ explanation 2]]

In this lesson, we'll learn to:

  1. Solve linear equations
  2. Solve linear inequalities
  3. 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

[[☃ video 1]]

###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 $2x+1=5$, what is the value of $x$ ?

[[☃ explanation 3]]

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

When combining like terms, recall that:

$ax\pm bx=(a \pm b)x$

Example: If $2x-4=5-x$, what is the value of $x$ ?

[[☃ explanation 4]]

When distributing coefficients, recall that:

$a(bx\pm c)=abx\pm ac$

Example: If $2(x+1)=5$, what is the value of $x$ ?

[[☃ explanation 5]]

####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 $\dfrac{1}{2}x+3=5$, what is the value of $x$ ?

[[☃ explanation 6]]

  • 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 $\dfrac{1}{2}x+\dfrac{1}{3}=\dfrac{1}{5}$ ?

[[☃ explanation 7]]

When working with negative numbers, remember that:

  • $\text{negative}\cdot \text{negative}=\text{positive}$
  • $\text{positive}\cdot \text{negative}=\text{negative}$

Example: If $-2(x-5)=1$, what is the value of $x$ ?

[[☃ explanation 8]]

####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 $2x+5y=1$ and $x=3$, what is the value of $y$ ?

[[☃ explanation 9]]

####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:

  1. Solve the linear equation, then plug the value of the variable into the expression to evaluate it.
  2. 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 $2x+1=5$, what is the value of $8x+4$ ?

[[☃ explanation 10]]

###Try it!

[[☃ graded-group 1]]

##How do I solve linear inequalities?

###Multi-step inequalities

[[☃ video 2]]

###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 $2x+1>5$ ?

[[☃ explanation 11]]

####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 $-2x+1>5$ ?

[[☃ explanation 12]]

###Try it!

[[☃ graded-group 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

[[☃ video 3]]

###Creating an equation with infinitely many solutions

[[☃ video 4]]

###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!

$2x-1=ax-2$

If $a=2$ in the equation above, what value of $x$ satisfies the equation?

[[☃ explanation 13]]

$2x-4=a(x-2)$

If $a=2$ is the equation above, what value of $x$ satisfies the equation?

[[☃ explanation 14]]

###Try it!

[[☃ graded-group 3]]

##Your turn!

[[☃ graded-group 4]]

[[☃ graded-group 5]]

[[☃ graded-group 6]]

[[☃ graded-group 7]]

##Things to remember

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.

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!)