Solve for x

Solve for an unknown: Linear Equations

Find all values of the unknown quantity that make the equation true.

When asked to solve for an unknown quantity in a linear equation, we attempt to find the values of that variable that make the equation true.

Example 1: Solve for $$x$$.

\[7+x=12\]

In this case, we need to find the value or values of x that make the equation true. For an equation to be true, the left hand side must be equivalent to the right hand side. Thus, by inspection, we see that the only way for $$7+x$$ to equal 12 is if $$x$$ equals 5.

In the following problems, find the value or values of the unknown that make the equation true. Don’t think about the steps of solving an equation. Instead, think about finding the value of the unknown that makes the equation true without writing anything down.

1. $$14-y=9$$

2. $$4x=28$$

3. $$y \cdot 72=9$$

4. $$7x+1=50$$

5. $$32y-2=6$$

Check your answers:

1. $$y=5$$
2. $$x=7$$
3. $$y={\large\frac{1}{8}}$$
4. $$x=7$$
5. $$y={\large\frac{1}{4}}$$

Sometimes, it is not possible to solve a linear equation using mental calculations. However, even when writing steps to solve an equation, it is important to remember that the intent of solving that equation is to find all of the values of the unknown quantity that make the equation true.

One way to approach this type of problem is to determine the operations acted on the unknown quantity and then think about how to undo (reverse) these operations. For instance, if a number is added to x, we need to subtract that number from the side of the equation that contains x. However, whenever an operation is performed on one side of an equation, it must also be performed on the other side. This is because the equal sign indicates that both sides of the equation have the same value. So, if we change the value of one side, we must change the value of the other side by the same amount to maintain that equality.

Example 2: Solve for x. That is, find the value(s) of x that make the equation true.

\[{\frac{2x}{15}}+21+{\frac{x}{5}}=28\]

Often, a good strategy when solving equations is to first simplify both sides of the equation. In this example, we can combinine like terms on the left hand side since there are two terms that contain an x s added together. Note that $${\Large\frac{2x}{15}}={\Large\frac{2}{15}}x$$ and similarly, $${\Large\frac{x}{5}}={\Large\frac{1}{5}}x$$. Thus, the equation can be written:

\[{\frac{2}{15}}x+21+{\frac{1}{5}}x=28\]

Then, by combining like terms, we get:

\[\left({\frac{2}{15}}+{\frac{1}{5}}\right)x+21=28\]

which is equivalent to: $${\Large\frac{1}{3}}x+21=28$$ (Click for review on adding fractions).

Thus, in order to solve for x, we need to undo the operations that have been done to it.

It is generally best to undo the operations in the opposite order that you perform them. For instance, on the left hand side of this equation, according to the order of operations, we first multiply x by $${\large\frac{1}{3}}$$ and then add 21 to it. So, in reverse order, as we undo what has been done to x, we first subtract 21 from each side:

\[{\frac{1}{3}}x+21-21=28-21\]

\[{\frac{1}{3}}x=7\]

and then divide by $${\Large\frac{1}{3}}$$ (which is the same as multiplying by 3) to undo the operation of multiplying x by $${\Large\frac{1}{3}}$$:

\[3 \cdot {\frac{1}{3}}x=3 \cdot 7\]

and now we have:

\[x=21\]

It is always a good idea to check your answer. We can check the answer to this linear equation by replacing each x in the original equation with 21. If this value of x is a solution, then the original equation will be true.

\[{\frac{2x}{15}}+21+{\frac{x}{5}}={\frac{2 \cdot 21}{15}}+21+{\frac{21}{5}}\]

Simplifying and adding fractions, the result is:

\[={\frac{14}{5}}+21+{\frac{21}{5}}\]

After simplifying the fractions, we get:

\[\eqalign{&=7+21\\ &=28}\]

Thus, since the value of 21 for x resulted in 28 on the left hand side, which is equal to the right hand side, the equation is true. Thus, the solution $$x=21$$ is correct.

Example 3: Solve for y. That is, find the values of y that make the equation true.

\[my+3n-c=5n+3c\]

First, the intent is once again to find the values of y that make the equation true. Although there are other variables in the equation, the process is still the same. We need to undo the things that have been done to y in order to isolate it and find its value in terms of the other variables.

In terms of order of operations, the last thing that is done to y on the left hand side is that c is subtracted from it. So, undo this first by adding c to each side:

\[my+3n−c+c=5n+3c+c\]

Then, combining like terms:

\[my+3n=5n+4c\]

Next, we see that $$3n$$ was added to the term containing the y, so now subtract $$3n$$ from both sides:

\[my+3n−3n=5n+4c−3n\]

After, combining like terms, we get:

\[my=2n+4c\]

Now, the only thing left that has been done to y is that it was multiplied by m. Here, we must be careful. Division is the operation that will undo multiplication, so we would like to divide by m. However, m is a variable and, thus, could be any number. Recall that you cannot divide by zero. Thus, we can divide by the m as long as it is not zero. So, assuming that m is not zero, we divide both sides by m, and get:

\[{\frac{my}{m}}={\frac{2n+4c}{m}}\]

And so, our final answer is:

\[y={\frac{2n+4c}{m}}\rm{\;,\; when\;} m \ne 0 \]

Some practice problems to check your skills:

1. Solve for x. That is, find the values of x that make the equation true:

$$3(x + 5) = 6$$

Think about it for a moment and then access this link to view answer.

2. Solve for y. That is, find the values of y that make the equation true:

\[y - (6y + 9) = 2y + 6\]

Think about it for a moment and then access this link to view answer.

3. Solve for x. That is, find the values of x that make the equation true:

\[2x + 3 - 4x = {{x \over 4}}\]

Think about it for a moment and then access this link to view answer.

4. Solve for a. That is, find the values of a that make the equation true:

\[{{{15a + 4} \over b}} = 12v - 5t\rm{\;,\; with\;} b \ne 0\]

Think about it for a moment and then access this link to view answer.

5. Solve for m. That is, find the values of m that make the equation true:

\[am - c = 7b + 2c\]

Think about it for a moment and then access this link to view answer.