1) Why should we clear fractions when solving linear equations and inequalities? Demonstrate how this is done with an example. Why should we clear decimals when solving linear equations and inequalities? Demonstrate how this is done with an example.
It makes it easier to walk through.
We clear fractions when solving linear equations and inequalities to make the process simpler and eliminate any potential errors. Fractions can make equations and inequalities more complicated to work with, so by clearing them, we can work with whole numbers instead.
To clear fractions in an equation or inequality, we can follow these general steps:
1) Identify all the fractions in the equation or inequality.
2) Find the least common multiple (LCM) of the denominators of the fractions.
3) Multiply every term in the equation or inequality by the LCM of the denominators.
4) Simplify the resulting equation or inequality by distributing and combining like terms.
5) Solve the simplified equation or inequality as you would normally.
Let's demonstrate this with an example:
Example 1: Solve the equation (2/3)x - 1/4 = 5/6
Step 1: Identify the fraction in the equation: 2/3
Step 2: Find the LCM of the denominator 3 and 4, which is 12.
Step 3: Multiply every term by 12. We have:
12 * (2/3)x - 12 * (1/4) = 12 * (5/6)
8x - 3 = 10
Step 4: Simplify the equation:
8x - 3 = 10
Step 5: Solve the simplified equation:
8x = 10 + 3
8x = 13
x = 13/8
Therefore, the solution to the equation (2/3)x - 1/4 = 5/6 is x = 13/8.
Similarly, we clear decimals in linear equations and inequalities for the same reasons. Decimals can make calculations more complicated and can lead to rounding errors. By clearing decimals and working with whole numbers, we can ensure more accurate results.
To clear decimals in an equation or inequality, we can multiply every term by a power of 10 that moves the decimal point to the right until we have whole numbers.
Let's demonstrate this with an example:
Example 2: Solve the equation 0.5x + 0.25 = 1.75
Step 1: Identify the decimal in the equation: 0.5
Step 2: Multiply every term by a power of 10 to clear the decimal. In this case, we can multiply every term by 10:
10 * (0.5x) + 10 * (0.25) = 10 * (1.75)
5x + 2.5 = 17.5
Step 3: Simplify the equation:
5x + 2.5 = 17.5
Step 4: Solve the simplified equation:
5x = 17.5 - 2.5
5x = 15
x = 15/5
x = 3
Therefore, the solution to the equation 0.5x + 0.25 = 1.75 is x = 3.