Problem 32 - Algebra for College Students

Question

Solve each equation for \(y\). $$-5 x+3 y=12$$

Step-by-Step Solution

Answer

y = 4 + \frac{5x}{3}

Step 1 - Isolate the variable involving y

Start by isolating the term that includes the variable y. To do this, add 5x to both sides of the equation:\[-5x + 3y = 12\]\[3y = 12 + 5x\]

Step 2 - Solve for y

Next, solve for y by dividing both sides of the equation by the coefficient of y, which is 3:\[y = \frac{12 + 5x}{3}\]Or equivalently,\[y = 4 + \frac{5x}{3}\]

Key Concepts

Isolate the Variable

To solve linear equations, it's often helpful to isolate the variable you want to solve for. In our case, we need to isolate the term involving y. This means we need to move all the other terms to the opposite side of the equation. Here’s how:

• Identify the term with the variable y. In the equation \(-5x + 3y = 12\), it's the term \(3y\).
• Move the other terms to the opposite side by performing the inverse operation. Since \(-5x\) is subtracted from \(3y\), we add \5x\ to both sides.

This leaves us with the equation \(3y = 12 + 5x\).
Isolating the variable makes it easier to solve for y in the next steps.

Solve for y

Once we have isolated the variable y on one side, our main goal is to solve for y. This requires us to eliminate any coefficients attached to y by performing the correct operations.
Here’s what we do next:

• We see that \(3y\) means y is multiplied by 3. Hence, to solve for y, we divide both sides of the equation by 3.

When we perform the division, we find \(y = \frac{12 + 5x}{3}\). Alternatively, we can split the fraction into two parts: \(y = 4 + \frac{5x}{3}\).
Both forms are correct and show that y is expressed in terms of x.

Understanding Linear Equations

Linear equations form a crucial part of algebra and can be in the form \(ax + by = c\). The equation \(-5x + 3y = 12 \) is a linear equation.
Linear equations can always be graphed as straight lines on a coordinate plane.

• The term \ \(ax + by\) is known as the linear expression.
• The constants a, b, and c in the equation are coefficients.

In our example, to solve for y, we transformed the equation into the slope-intercept form \(\boldsymbol{y = mx + b}\boldsymbol)\), which shows the relationship between y and x clearly.
This understanding helps us predict how changes in x will affect y and is vital for graphing purposes.