Question

Solve and check each equation. Treat the constants in these equations as exact numbers. Leave your answers in fractional, rather than decimal, form. $$44-11 p+4 p=-5$$

Short answer

The solution to the equation \(44 - 11p + 4p = -5\) is \(p = 7\). The solution is confirmed to be correct upon checking.

Step-by-step solution

Combine like terms on the left side of the equation

Start by combining the terms that contain the variable `p`. The given equation is \(44 - 11p + 4p = -5\). Combine the \(p\) terms as follows: \(-11p + 4p = -7p\). So, the simplified equation is \(44 - 7p = -5\).

Isolate the variable term

Subtract 44 from both sides to isolate the term containing the variable. The equation then becomes \(-7p = -5 - 44\), which simplifies to \(-7p = -49\).

Solve for the variable

Divide both sides by \(-7\) to solve for \(p\). This gives \(p = \frac{-49}{-7}\). Simplifying the fraction, we get \(p = 7\).

Check the solution

Substitute \(p = 7\) back into the original equation to verify the solution. The left-hand side of the original equation becomes \(44 - 11(7) + 4(7) = 44 - 77 + 28 = -33 + 28 = -5\), which matches the right-hand side of the original equation. Thus, the solution checks out.

Key concepts

Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and the first power of a single variable. Linear equations are foundational in algebra and represent a straight line when graphed on a coordinate plane. Solving linear equations involves finding the value of the variable that makes the equation true.

For example, in the equation \(44 - 11p + 4p = -5\), the goal is to determine the value of \(p\) that satisfies the equation. The process of solving this involves several steps that simplify the equation and isolate the variable, which includes combining like terms and balancing the equation by performing inverse operations.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both contain algebraic expressions. Working with algebraic fractions involves the same principles as working with numerical fractions, such as finding a common denominator or simplifying fractions. In our example, the final step of solving for \(p\) results in an algebraic fraction \(\frac{-49}{-7}\), which simplifies to \(7\). Simplification is an important skill when dealing with algebraic fractions. It requires factoring numerators and denominators, reducing them to their lowest terms, and sometimes performing operations like addition, subtraction, multiplication, or division on fractions.

Variable Isolation

Variable isolation is a critical concept in algebra. It refers to the process of manipulating an equation to get the variable on one side by itself. This isolation allows for the solution to be clearly identified. In the given equation, we isolate the variable \(p\) by performing operations that undo the addition and multiplication affecting \(p\).

These operations must maintain the balance of the equation, which means whatever is done to one side must also be done to the other. For instance, subtracting \(44\) from both sides of our equation and then dividing both sides by \(-7\) isolates \(p\) and allows us to find its value.

Combining Like Terms

Combining like terms is a method used to simplify an algebraic expression or equation. Like terms are terms that have exactly the same variable parts raised to the same power. Only the coefficients of these like terms can differ. For the equation in our exercise, \(44 - 11p + 4p = -5\), we combine the terms involving \(p\), since they are like terms.

By calculating \(-11p + 4p\), we simplify the left side of our equation to \(44 - 7p\), hence making the equation less complex and easier to solve. Understanding how to combine like terms efficiently is essential for solving linear equations and other algebraic equations.