Math · College algebra · Worked example

Solve a linear equation whose x terms cancel

Solve $4 (x - 1) + 2 = 4 x + 3$ and $4 (x - 1) + 2 = 4 x - 2$ . How many solutions does each have?

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

Start with the first equation

The two equations share the same left side and differ only in the number on the right. Solve the first one with the usual steps.

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

Expand the bracket

The $4$ multiplies both terms inside the bracket, so $4 (x - 1)$ is $4 x - 4$ .

4 x - 4 + 2 = 4 x + 3

Combine the constants

$- 4 + 2 = - 2$ , so the left side is $4 x - 2$ . Both sides now have the same $x$ term, $4 x$ .

4 x - 2 = 4 x + 3

Subtract $4 x$ from both sides

Subtracting the same quantity from both sides is always allowed. Here it removes every $x$ , and what remains is $- 2 = 3$ .

\begin{array}{l} 4 x - 2 = 4 x + 3 \\ - 2 = 3 \end{array}

Read the last line

$- 2 = 3$ is false, and it does not depend on $x$ . Every line was equivalent to the one before it, so the original equation is false for every $x$ as well. It has no solution. The answer is not $x = 0$ , because $x = 0$ gives $- 2 = 3$ too.

Solve the second equation

The same steps give $4 x - 2 = 4 x - 2$ . Subtracting $4 x$ from both sides leaves $- 2 = - 2$ , which is $0 = 0$ after adding $2$ . This last line is true whatever $x$ is, so every real number is a solution. The equation is an identity.

\begin{array}{l} 4 x - 2 = 4 x - 2 \\ 0 = 0 \end{array}

See it on a graph

Draw $y = 4 x - 2$ and $y = 4 x + 3$ . Both have slope $4$ , so they are parallel lines $5$ apart and never meet. That is why the first equation has no solution. For the second equation, both sides are the same line, and they meet at every point.

Result

$4 (x - 1) + 2 = 4 x + 3$ has no solution. $4 (x - 1) + 2 = 4 x - 2$ is an identity, true for every real number $x$ .

\begin{array}{l} no solution \\ every real x \end{array}

Your turn

1. One like the example

Solve $3 (2 x + 1) = 6 x + 7$ .

Show the answer and explanation

No solution.

Expanding gives $6 x + 3 = 6 x + 7$ . Subtracting $6 x$ from both sides leaves $3 = 7$ , which is false for every $x$ . The equation has no solution.

\begin{array}{l} 3 (2 x + 1) = 6 x + 7 \\ 6 x + 3 = 6 x + 7 \\ 3 = 7 \end{array}

2. A variation

Solve $5 x - 2 (x - 4) = 3 x + 8$ .

Show the answer and explanation

Every real number is a solution.

The $- 2$ multiplies both terms, so the left side is $5 x - 2 x + 8 = 3 x + 8$ . The equation becomes $3 x + 8 = 3 x + 8$ , and subtracting $3 x + 8$ from both sides leaves $0 = 0$ . It is an identity.

\begin{array}{l} 5 x - 2 (x - 4) = 3 x + 8 \\ 3 x + 8 = 3 x + 8 \\ 0 = 0 \end{array}

3. Somewhere new

Plan A costs $$ 20$ a month plus $$ 0.05$ a text. Plan B costs $$ 15$ a month plus $$ 0.05$ a text. For how many texts do the plans cost the same?

Show the answer and explanation

For no number of texts. Plan B is always $$ 5$ cheaper.

Setting the costs equal gives $20 + 0.05 t = 15 + 0.05 t$ . Subtracting $0.05 t$ leaves $20 = 15$ , which is false. The cost lines have the same slope and different starting values, so they are parallel and never meet.

\begin{array}{l} 20 + 0.05 t = 15 + 0.05 t \\ 20 = 15 \end{array}

Keep exploring

Open the steps in Math, then write $x = 0$ as a last row under $- 2 = 3$ . The Checker marks it ✗, because $x = 0$ does not solve the equation.

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After reading

Study it, then try it

Practice the key terms as flashcards, then open the example in its tool, change a value and keep a useful result on your board.

Study the 6 key terms Check the steps in Math Graph both sides in Graph Open worked example on a board Isolating a variable in Math Reference

Your existing work stays on this device. Examples open as editable copies.

Start with the first equation

Expand the bracket

Combine the constants

Subtract 4⁢x4x from both sides

Read the last line

Solve the second equation

See it on a graph

Result

Your turn

1. One like the example

No solution.

2. A variation

Every real number is a solution.

3. Somewhere new

For no number of texts. Plan B is always $5$5 cheaper.

Keep exploring

Study it, then try it

Subtract $4 x$ from both sides

For no number of texts. Plan B is always $$ 5$ cheaper.