Solve 8p When 2p = 3 (Basic Algebra Steps)

Given (2p=3), the value of (8p) is (12). Since (8) is four times (2), multiply both sides of the equation by (4): (4(2p)=4(3)). This gives (8p=12). The key is to apply the same operation to both sides so the equation stays balanced.

It is easy to find (p=\frac{3}{2}) and stop there, especially if you are used to solving equations for a variable. But here, the question asks for (8p), not just (p). The quickest path is to compare the coefficient of (p) in the given equation with the coefficient in the expression you need to find.

Problem Setup & Algebraic Principle

An equation says that two expressions have the same value. In (2p=3), the expression (2p) equals (3). To find (8p), notice that (8p) is four times (2p), so multiply the entire equation by (4) and keep both sides equal.

The letter (p) stands for a number. A number written beside a letter means multiplication, so (2p) means (2\times p), while (8p) means (8\times p). It does not mean (8+p).

The goal is not to find (p) first unless that helps you check your work. Instead, use the information already given:

• Given: (2p=3)
• Target: (8p)
• Coefficients: (8) is four times (2)

An equation remains true when you multiply both sides by the same number. Because (8p=4(2p)), multiplying (2p=3) by (4) gives the value of the target directly.

Why multiplying both sides works

Multiplying both sides by the same number preserves equality. If two quantities are equal, four times one quantity is equal to four times the other. Here, four times (2p) becomes (8p), and four times (3) becomes (12).

This rule is sometimes called the multiplication property of equality. It helps you change an equation without changing whether it is true. The number (4) is a valid multiplier here: it changes (2p) into the expression you want, (8p).

You can think of the equation as a statement that both sides have equal value. Changing only one side would make a different statement. Changing both sides in the same way preserves the original relationship.

Equation Steps & Required Operations

To find (8p), first compare its coefficient, (8), with the coefficient (2) in the given expression. Since (8=4\times2), multiply the left and right sides of (2p=3) by (4). Then simplify each side to get (8p=12).

Write the steps in order:

[ \begin{aligned} 2p&=3\ 4(2p)&=4(3)\ 8p&=12 \end{aligned} ]

The left side simplifies because (4\times2p=8p). The right side simplifies because (4\times3=12). Therefore, the requested value is:

[ \boxed{8p=12} ]

How to choose the multiplier

The multiplier connects the given expression to the target expression. Divide the target coefficient by the given coefficient:

[ \frac{8}{2}=4 ]

That tells you to multiply the entire equation by (4). This is a useful method when a problem gives one multiple of a variable and asks for another multiple of the same variable.

For the question at hand, the first row applies. The calculation is (4(2p)=4(3)), which simplifies to (8p=12). The other rows show the same pattern, not different rules.

Verification & Common Errors

A check confirms that the result agrees with the original equation. Solve the given equation for (p), then substitute that value into (8p). The check gives (p=\frac32) and (8p=8\left(\frac32\right)=12), matching the answer from multiplying both sides by (4).

To solve (2p=3) for (p), divide both sides by (2):

[ p=\frac{3}{2} ]

Then evaluate the requested expression:

[ 8p=8\left(\frac{3}{2}\right)=\frac{24}{2}=12 ]

Both methods give (12). The direct method is shorter because it finds (8p) without first calculating (p). The substitution method is useful as a check.

Common mistakes to avoid

A common mistake is to multiply only the left side:

[ 2p=3 \quad\not\Rightarrow\quad 4(2p)=3 ]

That changes only one side, so the new statement does not preserve the original equality. If you multiply the left side by (4), multiply the right side by (4) as well.

Another mistake is to answer (p=\frac32) without calculating (8p). That value of (p) is correct, but it is not the requested expression. Multiply (\frac32) by (8) to get (12).

A fast final check is to ask two things: Did you apply the same operation to both sides? Did you give the value of (8p), rather than only the value of (p)? If both answers are yes, your method matches the question.

Application to Similar Problems

The same method works whenever a problem gives a value for one multiple of a variable and asks for another. Compare the two coefficients, use their ratio as a multiplier, and apply it to both sides. Then simplify the result and check that it answers the exact expression requested.

For example, if (3x=10) and the target is (6x), the target coefficient (6) is twice the given coefficient (3). Multiply both sides by (2):

[ 2(3x)=2(10) ]

So (6x=20). This uses the same steps as the (p) problem.

If the target coefficient is smaller, the multiplier can be a fraction. For example, from (4q=12), to find (2q), use (2\div4=\frac12). Multiplying both sides by (\frac12) gives (2q=6). A fraction multiplier is not a different rule; it still applies the same operation to both sides.

A repeatable checklist

Use this short checklist for problems of this kind:

• Identify the given equation and the expression the question asks you to find.
• Compare the target coefficient with the given coefficient.
• Divide the target coefficient by the given coefficient to find the multiplier.
• Multiply both sides by that multiplier.
• Simplify, then make sure your final answer names the requested expression.

For this example, the multiplier is (8\div2=4). Applying it gives (4(2p)=4(3)), so (8p=12). The important idea is not to memorize “multiply by four” for every problem. The multiplier depends on the coefficients in the particular question.

Conclusion

When (2p=3), the expression (8p) is four times (2p), because (8=4\times2). Multiply both sides of the given equation by (4), simplify, and get (8p=12). Always preserve equality by applying the same operation to both sides, and make sure your final answer matches what the question asks.

Frequently Asked Questions

What is (8p) if (2p=3)?

(8p=12). Multiply both sides of (2p=3) by (4), because (8) is four times (2).

What is the first step?

Compare the coefficients of (p): the given coefficient is (2), and the target coefficient is (8). Since (8\div2=4), multiply the equation by (4).

Why do I multiply by (4)?

Multiplying (2p) by (4) gives (8p). To keep the equation true, multiply the other side, (3), by (4) as well.

Should I solve for (p) first?

You can, but you do not have to. Solving gives (p=\frac32), and then (8p=12). Multiplying the original equation by (4) is a shorter route.

What is the value of (p)?

The value is (p=\frac32), or (1.5). That is an intermediate result if the question asks for (8p); the requested value is (12).

Does (8p) mean (8+p)?

No. A number beside a variable means multiplication, so (8p) means (8\times p). Addition would be written as (8+p).

Can I multiply only the left side by (4)?

No. You must multiply both sides by (4). Changing only one side does not preserve the equation.

How can I check that (12) is correct?

Use (p=\frac32), found from (2p=3). Then (8p=8\times\frac32=12), which confirms the result.

Could the multiplier be a fraction in another problem?

Yes. If the target coefficient is smaller than the given coefficient, their ratio may be a fraction. Apply that same fraction to both sides of the equation.

What is the main rule to remember?

To change one multiple of a variable into another, compare their coefficients and use the same multiplier on both sides. Then simplify and answer the expression the question names.

(This article was written by one of our staff writers, Alan Westbrook. Visit our Meet the Team page.)