Problem 64 - Prealgebra

Question

Use the formula \(2 x+5 y=10\) to find \(y\) if: $$x=-5$$

Step-by-Step Solution

Answer

When \(x = -5\), then \(y = 4\).

Step 1: Substitute the x-value

First, substitute the given value of \(x\) into the equation. You replace \(x\) with -5 in the equation \(2x + 5y = 10\). This gives you the equation \(2(-5) + 5y = 10\).

Step 2: Simplify the equation

Next, simplify the equation by calculating \(2(-5)\), which is -10. So, the equation now becomes \(-10 + 5y = 10\).

Step 3: Isolate the term with y

Now, to isolate the \(5y\) term, add 10 to both sides of the equation. This simplifies the equation to \(5y = 20\).

Step 4: Solve for y

Finally, solve for \(y\) by dividing both sides of the equation by 5. Thus, the equation becomes \(y = \frac{20}{5}\), which simplifies to \(y = 4\).

Key Concepts

Substitution method

The substitution method is an effective technique often used to solve linear equations like the one in this exercise. It involves replacing a variable with a given value to simplify the equation. In the original problem, we have the equation \(2x + 5y = 10\) and are given that \(x = -5\). We substitute \(-5\) for \(x\) in the equation. This gives us \(2(-5) + 5y = 10\). By doing this substitution, we transform the equation to eliminate \(x\) entirely, focusing our efforts solely on solving for \(y\).

This method is particularly useful when one variable's value is already known, allowing you to resolve equations that otherwise might seem complex. The substitution simplifies the process by reducing the unknowns, leading to more straightforward calculations and solutions.

Isolation of variables

The isolation of variables is a crucial step in solving equations, especially when aiming to find the value of a specific variable. After substituting the given \(x\)-value into the equation, the rewritten equation becomes \(-10 + 5y = 10\). The goal is to isolate \(y\), allowing us to solve for it independently.

To achieve isolation, we begin by moving all terms that do not contain \(y\) to the other side of the equation. In this case, adding 10 to both sides simplifies it to \(5y = 20\). Through isolation, we have a much clearer path to solving the equation, focusing solely on the variable \(y\). This step is critical because it reduces the equation to a simpler form, where standard arithmetic operations can easily yield the solution.

Simplification of equations

Simplification is a fundamental aspect of solving equations, ensuring they are easier to work with. After substituting and rearranging the original equation to \(-10 + 5y = 10\), the next step is simplification.

First, by simplifying \(-10\) (the product of \(2\) and \(-5\)) and moving it to the other side of the equation, we get \(5y = 20\). This equation is significantly simpler.

Now, to finalize the solution for \(y\), divide every term by \(5\), resulting in \(y = \frac{20}{5}\), simplifying further to \(y = 4\). Simplifying equations is all about making complex expressions simple enough to solve using basic arithmetic, which leads to clear and easily understandable answers.