Problem 42 - Algebra 1

Question

Multiply the equation by a power of 10 to write an equivalent equation with integer coefficients. $$ 1.67+2.43 x=3.29(x-5) $$

Step-by-Step Solution

Answer

The equivalent equation with integer coefficients is \(167 + 243x = 329x - 1645\).

Step 1: Multiply Each Term by 10^2

To write an equivalent equation with integer coefficients, multiply each term in the equation by 100. \(100 \times 1.67 + 100 \times 2.43x = 100 \times 3.29(x-5)\).

Step 2: Simplify Each Term

By doing the multiplication, we get \(167 + 243x = 329(x - 5)\). This equation is equivalent to the original one, but has integer coefficients.

Step 3: Distribute On the Right Hand Side

Now, distribute 329 on the right-hand side of the equation to get \(167 + 243x = 329x - 1645\).

Step 4: Write The Conclusion

Final representation of the equation with integer coefficients is \(167 + 243x = 329x - 1645\). This equivalent equation has integer coefficients, as required.

Key Concepts

Multiplying by Powers of Ten

When working with equations that have decimal numbers, such as 1.67 or 2.43, converting them to integers can make handling them much simpler. This is done by multiplying each term in the equation by the same power of ten. For example, multiplying by 10 moves the decimal one place to the right, by 100 moves it two places, and so on. In our exercise, we multiply each term by 100 to eliminate the decimals:

• \(100 \times 1.67 = 167\)
• \(100 \times 2.43x = 243x\)
• \(100 \times 3.29(x-5) = 329(x-5)\)

By doing this, the equation becomes easier to work with because all coefficients are now integers. Choosing the correct power of ten often depends on the number of decimal places you need to clear, and in this case, two decimal places lead us to multiply by 100.

Equivalent Equations

An equivalent equation is one that has the same solution as the original equation, although it might look different. By multiplying an equation by a nonzero number, we don't change its solutions, but only its appearance. In our exercise, the original equation \(1.67 + 2.43x = 3.29(x-5)\) becomes \(167 + 243x = 329(x-5)\) after multiplying by 100. This transformation doesn't alter the solutions of the equation; it just provides an alternate form without decimals. Always ensure that what you multiply by is a nonzero number since multiplying by zero will not preserve the original equation's essence.

Simplifying Equations

Once you have multiplied by a power of ten to eliminate decimals, you're tasked with simplifying the equation. Begin by carrying out any necessary distributions or combinations of like terms. In our example, we distribute 329 on the right-hand side:

• \(329(x - 5) = 329x - 1645\)

Now, the equation is \(167 + 243x = 329x - 1645\). Simplifying an equation involves putting it in the most reduced form while preserving equivalence with the original. This step often makes further problem-solving easier and clearer. Always verify that each step maintains the integrity of the equation's solutions.