Equations with Rational Numbers

Warm Up

You can solve equations with rational number coefficients and constants by first eliminating the fractions or decimals from the equation by multiplying both sides of the equation by the same factor. Continue by isolating the variable using the same steps that were used for solving equations with integer coefficients and constants.

Why must you multiply both sides of the equation by the LCM Why must you multiply both sides of the equation by the LCM? The same operations must be carried out on both sides of the equation so that the resulting equation has the same solution as the original equation.

When multiplying by the LCM all terms on the left and on the right of the equal sign get multiplied by the LCM, not just the fractions.

Sasha walks to school at a constant rate. She meets her friend Hannah 0.6 kilometers along the way, and they continue walking at the same constant rate. When they get to school, Sasha has walked for 0.6 hour and Hannah has walked for 0.45 hour. What is the rate in kilometers per hour at which they walked to school?

How do you determine what to multiply an equation by so that you eliminate fractions or decimals from the equation? If the equation has fractions in it, multiply both sides of the equation by the LCM (or any common multiple) of the denominators. If the equation has decimals in it, multiply both sides of the equation by the appropriate power of 10 to eliminate the decimals.

The solution of an equation can be a decimal or a fraction, even though you eliminated fractions or decimals from the original equation.

Write a real-world situation that can be modeled by the equation 5 Write a real-world situation that can be modeled by the equation 5.25x + 3.25 = 1.75x + 10.25

When eliminating decimals from an equation the greatest number of decimal places in the terms will determine the power of 10 you should multiply by. For an equation with terms having decimal numbers in the tenths, hundredths, and thousandths, you would multiply by 103 or 1000 since thousandths is the greatest decimal place value. The number of place values to the left of the decimal has no bearing on which power of 10 to use.

Exit Ticket 1. 7 8 𝑥 − 1 2 = 3 16 𝑥+5 2. Marsha is changing her phone service. One data package charges an initial fee of $50 and $13.25 per month. The other has no initial fee but charges $25.75 per month for the same data package. After how many months would Marsha have paid the same amount for either data package? 3. Write a real-world situation that can be modeled by the equation 8.35x = 4.25x + 36.90. 4. The perimeter of one square is given as 4x. Its perimeter is equal to the perimeter of a rectangle given as 2.5x + 3.75. What is the perimeter of each figure?