Solving Linear Equations, SAT math lesson

Picture a balance with equal weights on both pans. Add the same weight to each side and it stays level. Take the same weight off each side and it stays level too. Solving an equation is the same game played with numbers, and the rule is that whatever you do to one side you do to the other, using a step you could undo. Linear equations show up all over the SAT Math section, often inside word problems, so each step should become automatic and safe.

See also Solving Equations and Checking Solutions in Foundations for Calculus.

Key terms

Linear equation

Definition A linear equation in one variable $x$ is an equation whose two sides are linear expressions in $x$ , that is expressions that can be written as $ax + b$ with real constants $a$ and $b$ . The variable appears only to the first power, never multiplied by itself, never in a denominator and never under a root. Moving every term to one side turns such an equation into the form $ax + b = 0$ . When $a \ne 0$ it has exactly one solution,

x = -\frac{b}{a}.

When the variable cancels out completely, the equation has either no solution or infinitely many, which is the topic of Equations with No or Infinitely Many Solutions.

In plain words A linear equation has $x$ only in its plain form, possibly with a number in front. The graph of $y = ax + b$ is a straight line, which is where the name comes from.

Example The equations $5x - 4 = 2x + 5$ and $\frac{x}{3} + 1 = 7$ are linear. The equations $x^2 = 9$ and $\frac{1}{x} = 4$ are not, because $x$ is squared in the first and sits in a denominator in the second.

Solution

Definition A solution of an equation in $x$ is a number that makes the equation true when it is substituted for $x$ . The solution set is the set of all solutions. To check a candidate, substitute it into the original equation, not into a later line of your work, because a mistake in an early step carries into every later line.

In plain words A solution is a number that makes both sides come out equal. The original equation is the only fair judge.

Example Is $x = 3$ a solution of $5x - 4 = 2x + 5$ ? The left side is $5(3) - 4 = 11$ and the right side is $2(3) + 5 = 11$ . Both sides agree, so $x = 3$ is a solution. The value $x = 2$ is not, since the left side is $6$ and the right side is $9$ .

Equivalent equation

Definition Two equations are equivalent when they have exactly the same solution set. The following steps always produce an equivalent equation.

Replacing either side by an equivalent expression, for example by expanding or combining like terms.
Adding the same number to both sides, or subtracting it from both sides. The same holds for an expression such as $3x$ that has a value for every $x$ .
Multiplying or dividing both sides by the same nonzero number.

Each of these steps can be undone, so no solution is lost and none is added. Multiplying both sides by $0$ is not allowed, since it turns every equation into $0 = 0$ . Dividing both sides by an expression that contains the variable is also unsafe, since that expression may be zero for a solution. Never divide by zero.

In plain words A legal step is one you could reverse. If you can always get back to the original equation, you have neither gained nor lost a solution.

Example Look at $3x = 5x$ . Dividing both sides by $x$ gives $3 = 5$ , which suggests there is no solution. That is wrong, because $x = 0$ makes both sides $0$ . The division was illegal when $x = 0$ . The safe route subtracts $3x$ from both sides, giving $0 = 2x$ , and then divides by the nonzero number $2$ , giving $x = 0$ .

Inverse operation

Definition Two operations are inverse operations when one undoes the other. Adding $c$ and subtracting $c$ are inverses, and for $c \ne 0$ multiplying by $c$ and dividing by $c$ are inverses. To isolate a variable, apply the inverse of each operation that was done to it, to both sides of the equation, in the reverse of the order in which those operations were applied.

In plain words Peel the operations off the variable like layers, outermost layer first.

Example Solve $\frac{x}{4} - 3 = 2$ . The variable was divided by $4$ and then had $3$ subtracted, so undo the subtraction first,

\frac{x}{4} = 5,

and then multiply both sides by $4$ to get $x = 20$ . Check in the original. Since $\frac{20}{4} - 3 = 2$ , the solution is correct.

To clear fractions, multiply both sides by a common denominator, and keep every numerator with more than one term in parentheses. Solve

\frac{x}{3} + \frac{x - 1}{4} = 5.

Multiplying both sides by $12$ gives

\begin{aligned} 4x + 3(x - 1) &= 60 \\[0.4em] 7x - 3 &= 60 \\[0.4em] 7x &= 63, \end{aligned}

so $x = 9$ . The check gives $3 + 2 = 5$ . Writing $3x - 1$ instead of $3(x - 1)$ in the first line would lead to $x = \frac{61}{7}$ , which fails the check.

Distributive property

Definition For all real numbers $a$ , $b$ and $c$ ,

\begin{aligned} a(b + c) &= ab + ac, \\[0.4em] a(b - c) &= ab - ac. \end{aligned}

The factor in front multiplies every term inside the parentheses, and a minus sign in front of parentheses acts as a factor of $-1$ , so $-(b - c) = -b + c$ .

In plain words Hand the outside number to everyone inside. If it is negative, every sign inside changes.

Example Solve $3(2x - 1) = 2(x + 5) + 3$ , which has the variable on both sides. Expand both sides, collect the $x$ terms on one side and the numbers on the other,

\begin{aligned} 6x - 3 &= 2x + 10 + 3 \\[0.4em] 6x - 3 &= 2x + 13 \\[0.4em] 4x &= 16 \\[0.4em] x &= 4. \end{aligned}

Check in the original. The left side is $3(8 - 1) = 21$ and the right side is $2(9) + 3 = 21$ .

Requested expression

Definition Many questions ask for the value of an expression such as $2x$ , $x + 1$ or $3x - 2$ rather than for $x$ itself. This requested expression is what you must report. You can find it by solving for the variable and substituting, or often faster by multiplying, dividing, adding or subtracting the whole equation so that the requested expression appears alone on one side.

In plain words Read the last line of the question twice. If it asks for $3x - 2$ , giving the value of $x$ is a wrong answer, and it is often one of the choices.

Example If $6x - 4 = 20$ , what is the value of $3x - 2$ ? The left side is exactly twice $3x - 2$ , so dividing both sides by $2$ gives

3x - 2 = 10.

Solving first gives the same result, since $6x = 24$ gives $x = 4$ and then $3(4) - 2 = 10$ . The trap answer is $4$ .

Check yourself

What is the solution to the equation $4(x - 3) - 2x = 10$ ? (A) $-1$ (B) $6.5$ (C) $11$ (D) $22$

Show the answer

The correct choice is (C). Expanding gives $4x - 12 - 2x = 10$ , so $2x = 22$ and $x = 11$ . The check gives $4(8) - 22 = 10$ . Choice (A) comes from writing $+12$ instead of $-12$ , choice (B) from multiplying only the $x$ by $4$ , and choice (D) is the value of $2x$ , not $x$ .

If $5x + 3 = 2x + 21$ , what is the value of $2x$ ?

Show the answer

The answer is $12$ . Subtracting $2x$ and then $3$ from both sides gives $3x = 18$ , so $x = 6$ and $2x = 12$ . The check gives $33$ on both sides of the original. Entering $6$ answers a question that was not asked.

What value of $x$ satisfies the equation below?

\frac{x}{2} - \frac{x - 4}{6} = 3

Show the answer

The answer is $7$ . Multiplying both sides by $6$ gives $3x - (x - 4) = 18$ , which is $2x + 4 = 18$ , so $x = 7$ . The check gives $\frac{7}{2} - \frac{3}{6} = 3.5 - 0.5 = 3$ . Without the parentheses the minus sign would reach only the $x$ , giving $2x - 4 = 18$ and the wrong value $11$ .

Solving Linear Equations

Theory, part 13

Practice, part 14

Key terms

Linear equation

Solution

Equivalent equation

Inverse operation

Distributive property

Requested expression

Check yourself

How this lesson connects

Builds on

Used later in