Problem 63 - Prealgebra

Question

Use the formula \(2 x+5 y=10\) to find \(y\) if: $$x=0$$

Step-by-Step Solution

Answer

When \(x=0\), \(y=2\).

Step 1: Substitute the Given Value

We need to find the value of \(y\) when \(x=0\). Start by substituting \(x=0\) into the equation \(2x + 5y = 10\). This will give us a new equation: \(2(0) + 5y = 10\).

Step 2: Simplify the Equation

Calculate \(2(0)\), which equals 0. This simplifies the equation to \(5y = 10\), since \(2 \times 0 = 0\) does not contribute to the sum.

Step 3: Solve for \(y\)

We have the equation \(5y = 10\). To find \(y\), divide both sides of the equation by 5. This gives \(y = \frac{10}{5}\).

Step 4: Calculate the Final Value

Perform the division \(\frac{10}{5}\) to get \(y = 2\). This is the value of \(y\) when \(x=0\).

Key Concepts

Substitution Method

The substitution method is a powerful tool for solving systems of linear equations. It involves replacing one variable in an equation with an expression derived from another equation. This method simplifies the process and makes it possible to solve for one variable at a time, which can be particularly useful when you have specific values given.
Let's consider our exercise, where we have the equation \(2x + 5y = 10\), and we're given \(x = 0\). By substituting \(x = 0\) into the equation, we simplify the equation to a single variable, making it easier to solve.

• First, identify which variable you can substitute by seeing what values are given or what equations might be the easiest to work with.
• Substitute the known value into the equation, just as we did by replacing \(x\) with 0 to obtain \(5y = 10\).

Once substitution simplifies your equation to a single variable, you are on the path to finding a straightforward solution.

Solving Equations

Solving equations is the process of finding the value of unknown variables that make the equation true. In our example, after substituting \(x = 0\) into \(2x + 5y = 10\), we simplified our equation to \(5y = 10\).
To solve this equation, follow these clear steps:

• Isolate the variable: Since the equation \(5y = 10\) is already in a good form, we need to make \(y\) the subject.
• Divide both sides by the coefficient of \(y\): In this case, divide both sides by 5. This results in \(y = \frac{10}{5}\).
• Solve the division: Calculate \(\frac{10}{5}\), resulting in \(y = 2\).

Always check your solution by plugging it back into the original equation to ensure it satisfies all conditions. Here, when \(x = 0\) and \(y = 2\), the equation indeed holds true as \(10 = 10\).

Simplifying Expressions

Simplifying expressions is a key skill in algebra, crucial for making equations easier to handle. It involves reducing an expression to its simplest form while retaining equality.
In the context of our exercise, after substituting \(x = 0\) in the equation, we simplify \(2(0) + 5y = 10\) to \(5y = 10\). The simplification process typically involves:

• Performing any operations necessary, such as multiplication or addition, as seen with \(2(0)\) turning to 0.
• Combining like terms if possible, although in this example, it's a direct simplification.
• Ensuring the equation remains balanced: Each operation done on one side of the equation should be mirrored on the other side to maintain equality.

Simplifying expressions might seem like a small step, but it significantly eases the process of finding solutions and allows for a clearer pathway to understanding the equation's structure.