Problem 20 - 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. $$7 x+15=8$$

Step-by-Step Solution

Answer

The solution to the equation \(7x+15=8\) is \(x = -1\).

Step 1: Isolate the variable term

Subtract 15 from both sides of the equation to get the term with the variable x by itself on one side. This gives us the simplified equation: \(7x = 8 - 15\).

Step 2: Combine constant terms

Combine the constants on the right-hand side of the equation. This entails subtracting 15 from 8, yielding \(7x = -7\).

Step 3: Divide both sides by coefficient of x

To solve for x, divide both sides of the equation by the coefficient of x, which is 7. This results in \(x = \frac{-7}{7}\).

Step 4: Simplify the fraction

Simplify the fraction on the right-hand side to find the value of x. \(x = -1\).

Step 5: Check the solution

Substitute x with -1 in the original equation to verify the solution: \(7(-1)+15=8\) simplifies to \(-7+15=8\), which is true.

Key Concepts

Isolating the Variable

In order to solve linear equations, such as the provided example of 7x + 15 = 8, the first objective is to isolate the variable. This means that we want the variable, in this case, x, to be on one side of the equation by itself. To achieve this, we perform operations that will eliminate other terms from the side of the equation where the variable is located.
Isolating a variable can involve several steps. Starting with - in our example - subtracting 15 from both sides of the equation, effectively balances the equation and leaves us with 7x isolated on one side. It is critical to keep the equation balanced by performing the same operation on both sides. When you perform these operations, imagine a set of scales, trying to keep them level as you add or remove weights (numbers or terms).

Combining Like Terms

Once the variable is isolated, the next step often involves 'combining like terms.' In the context of our exercise, combining like terms happened when we simplified the right-hand side of the equation after moving the constant term over. Because 8 - 15 are both constants, they can be combined into a single term, resulting in -7.
Like terms are terms in an equation that have the same variables raised to the same power, even if their coefficients are different. Only like terms can be combined through addition or subtraction. This process simplifies the equation and makes the next step—solving for the variable—straightforward. Remember to keep an eye out for signs (positive or negative) and to apply the correct operations accordingly to combine these terms accurately.

Fractional Solutions

Solving linear equations can often result in fractional solutions. In our example, after isolating the variable and combining like terms, we ended up with the equation 7x = -7. To find x, we divided both sides of the equation by 7, which is the coefficient of x. This division resulted in a fractional solution of x = -1, which is valid because the fraction -7/7 simplifies to -1.
It's crucial for students to understand that fractional solutions are acceptable answers in algebra and they should be comfortable working with them. Leaving answers in fractional form can be more precise than converting them to decimal form, especially when dealing with exact numbers rather than approximations. Always simplify fractions to their lowest terms, as this makes checking the solution in the original equation much more straightforward.