Problem 62 - Prealgebra

Question

Suppose \(4 x+3 y=12 .\) Find \(y\) if: $$x=-3$$

Step-by-Step Solution

Answer

When \( x = -3 \), \( y = 8 \).

Step 1: Substitute the Given Value of x

Start by substituting the given value of \( x = -3 \) into the equation \( 4x + 3y = 12 \). This gives us: \[ 4(-3) + 3y = 12 \] Simplify the expression by multiplying \( 4 \) and \( -3 \).

Step 2: Simplify the Equation

Simplifying \( 4(-3) \) gives us \(-12\), so the equation becomes: \[ -12 + 3y = 12 \] The next step is to isolate \( y \) by getting rid of the \( -12 \) on the left side.

Step 3: Add 12 to Both Sides

Add \( 12 \) to both sides of the equation to balance it: \[ -12 + 3y + 12 = 12 + 12 \] This simplifies to: \[ 3y = 24 \] Now \( y \) can be easily isolated by dividing both sides.

Step 4: Solve for y

Divide both sides of the equation by \( 3 \) to solve for \( y \): \[ y = \frac{24}{3} \] Simplifying the right side gives: \[ y = 8 \] This is the value of \( y \) when \( x = -3 \).

Key Concepts

Substitution Method

The substitution method is a way to solve systems or single linear equations by replacing a given value into one of the variables. By substituting the known value, the focus shifts to solving for the remaining variable. This technique simplifies the equation and makes it easier to reach a solution.

In our exercise, we were given the equation \(4x + 3y = 12\) and the value \(x = -3\). By substituting \(-3\) for \(x\), we transformed the equation into \(4(-3) + 3y = 12\). Using substitution reduces the number of variables, allowing us to concentrate on one variable at a time, which simplifies the problem-solving process.

The substitution method is especially useful in algebra, as it allows us to deal with fewer variables and solve for just one unknown variable at a time.

Isolation of Variable

Isolation of a variable is an essential step when solving equations. It means manipulating the equation to have one variable alone on one side of the equation. This is crucial for finding the values of variables.

In this case, after substituting \(x = -3\) into \( 4x + 3y = 12 \), our equation became \(-12 + 3y = 12\). To isolate \(y\), we had to eliminate the \(-12\) next to it. By adding \(12\) to both sides of the equation \(-12 + 3y = 12\), we achieved symmetry, transforming the equation to \(3y = 24\).

This process of isolation made it easy to then further solve for \(y\) by dividing both sides by the coefficient of \(y\), leaving \(y\) alone on one side. Isolation of variables is a powerful tool, making complicated equations much more manageable.

Simplifying Equations

Simplifying equations involves reducing them to their simplest form to make the problem-solving process more straightforward. It includes performing arithmetic operations and reducing terms where possible.

In our original equation, after substituting \(x = -3\), we had \(4(-3) + 3y = 12\). Simplifying this by performing the multiplication \(4 \times -3\) resulted in \(-12\). This turned the equation into \(-12 + 3y = 12\). Removing like terms or constants by performing operations like addition or subtraction simplifies further, as seen when adding 12 to both sides to get \(3y = 24\).

Finally, dividing the equation by the coefficient of the variable helps simplify the expression, resulting in \(y = 8\). These steps in simplifying maintain the balance of the equation while making it easier to isolate and solve for unknown variables efficiently.