Problem 11 - Technical Mathematics with Calculus

Question

Solve and check each equation. Treat the constants in these equations as exact numbers. Leave your answers in fractional, rather than decimal, form. $$10-5 y=1-y$$

Step-by-Step Solution

Answer

The solution to the equation is y = 9/4. The solution has been checked and is correct.

Step 1: Isolate the variable

To solve for the variable y, you should first collect all the y terms on one side of the equation. Do this by adding 5y to both sides of the equation to isolate the y terms on the right side, which gives you: 10 = 1 + 4y.

Step 2: Simplify both sides

Next, simplify the right side of the new equation by combining the like terms to get 10 = 1 + 4y.

Step 3: Isolate y completely

Subtract 1 from both sides of the equation to isolate the term with y, resulting in: 9 = 4y.

Step 4: Find the value of y

Now, divide both sides of the equation by 4 to solve for y, giving you y = 9/4.

Step 5: Check the solution

Substitute y with 9/4 back into the original equation to check if the left side equals the right side: 10 - 5(9/4) should equal 1 - (9/4). Simplify the left side to get 10 - 45/4, which simplifies to 40/4 - 45/4, resulting in -5/4. Simplify the right side to get 1 - 9/4, which simplifies to 4/4 - 9/4, also resulting in -5/4. Since the left side equals the right side after the substitution, the solution y = 9/4 is correct.

Key Concepts

Isolating Variables

The process of isolating variables is essential in solving linear equations. It involves rearranging the equation so that the variable you are solving for is on one side of the equation by itself. This is typically done by performing the same operation on both sides of the equation to maintain equality.

For instance, if you encounter an equation like \(10 - 5y = 1 - y\), your goal is to get \(y\) by itself on one side. You might start by adding \(5y\) to both sides to combine all the \(y\) terms together, resulting in the equation \(10 = 1 + 4y\). Following that, operations such as addition or subtraction are used to move constants to the opposite side, and finally multiplication or division is employed to solve for the variable.

Simplifying Equations

Simplifying equations is a process that makes equations easier to solve. It involves combining like terms and reducing fractions, among other operations, to obtain the simplest expression that's equivalent to the original. For example, in the equation \(10 = 1 + 4y\), simplifying would mean combining any like terms and reducing fractions on both sides, if any are present.

An important thing to remember is that whatever operation you perform on one side of the equation, you must also perform on the other side to keep the equation balanced. Keeping the equation in its simplest form not only makes it easier to understand but also reduces the likelihood of making mistakes when solving for the variable.

Fractional Solutions

When solving linear equations, solutions can sometimes be in the form of fractions, which represent exact values. For example, solving for \(y\) in the equation \(9 = 4y\) results in the fractional solution \(y = \frac{9}{4}\). It is important to be comfortable working with and simplifying fractions.

To check a fractional solution, substitute it back into the original equation and simplify both sides of the equation. If after the substitution, the two sides of the equation are equal, the solution is correct. In our case, inserting \(y = \frac{9}{4}\) back into the original equation \(10 - 5y = 1 - y\) gives you the same result on both sides, confirming the solution's validity. Being able to work with fractions is crucial, as they often appear in mathematical solutions and represent more precise values than decimals.