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Chapter 2: Problem 363

Solve each equation with decimal coefficients. $$1.2 x-0.91=0.8 x+2.29$$

Short Answer

Expert verified
x = 8

Step by step solution

01

Isolate terms involving x

Start by getting all terms involving x on one side of the equation. Subtract 0.8x from both sides: \( 1.2x - 0.91 - 0.8x = 0.8x + 2.29 - 0.8x \) This simplifies to: \( 0.4x - 0.91 = 2.29 \)
02

Isolate the variable term

Next, isolate the term containing x by adding 0.91 to both sides of the equation:\( 0.4x - 0.91 + 0.91 = 2.29 + 0.91 \)This simplifies to: \( 0.4x = 3.2 \)
03

Solve for x

Finally, solve for x by dividing both sides by 0.4:\( x = \frac{3.2}{0.4} \)This simplifies to: \( x = 8 \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Decimal Coefficients
When solving linear equations, you might encounter decimal coefficients. These are just like regular numbers but expressed in decimal form.
For example, in the problem given鈥�$$1.2x - 0.91 = 0.8x + 2.29$$鈥攖he coefficients of x are 1.2 and 0.8, and there are also decimal constants -0.91 and 2.29.
Don't let decimals intimidate you. Treat them the same way you would treat whole numbers. The goal remains the same: isolate the variable and solve the equation.
Isolate Variable
To find the value of the variable in an equation, you need to isolate it. This means getting the variable alone on one side of the equation. Let's look at the steps a bit more closely:
  • Move all terms involving the variable to one side of the equation.
  • Keep the constants (numbers without variables) on the other side.
  • Use basic arithmetic operations like addition, subtraction, multiplication, and division to achieve this.
In our example: $$1.2x - 0.91 = 0.8x + 2.29$$
We subtract 0.8x from both sides and get: $$0.4x - 0.91 = 2.29$$
Then, we add 0.91 to both sides to isolate 0.4x: $$0.4x = 3.2$$
Step-by-Step Solution
It's crucial to follow a clear, step-by-step approach when solving equations. Here鈥檚 a recap of the steps for our example:
  • First, we subtracted 0.8x from both sides to gather like terms: $$1.2x - 0.91 - 0.8x = 0.8x + 2.29 - 0.8x$$, which simplifies to: $$0.4x - 0.91 = 2.29$$
  • Next, we moved the constant -0.91 to the other side by adding 0.91: $$0.4x - 0.91 + 0.91 = 2.29 + 0.91$$, simplifying to: $$0.4x = 3.2$$
  • Finally, we divided by 0.4: $$x = \frac{3.2}{0.4}$$, giving us: $$x = 8$$
Breaking down each step ensures accuracy and clarity.
Simplifying Equations
Simplifying equations makes them easier to solve. Here are a few tips:
  • Combine like terms: Terms that have the same variable with the same exponent.
  • Clear decimals by multiplying the whole equation by a power of 10 if necessary.
  • Perform inverse operations to isolate the variable. For example, adding cancels out subtraction and vice versa.
In our example: $$1.2x - 0.91 = 0.8x + 2.29$$
We simplified by combining like terms and isolating the variable step-by-step,
getting: $$x = 8$$. Breaking down the equation into simpler parts makes it manageable and easier to solve.

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