Problem 31 - Algebra for College Students

Question

Solve each equation for \(y\). $$-3 x+2 y=5$$

Step-by-Step Solution

Answer

y = \frac{3x + 5}{2}

Step 1: Isolate the term with y

Start by adding \(3x\) to both sides of the equation to isolate the term with \(y\). This gives us: \[2y = 3x + 5\]

Step 2: Solve for y

To solve for \(y\), divide both sides of the equation by \(2\): \[y = \frac{3x + 5}{2}\]

Key Concepts

isolating variables

When solving a linear equation, one of the first steps is often to isolate the variable you're solving for. In our case, we need to isolate the term with the variable \(y\). Let's take our initial equation, \(-3x + 2y = 5\).
To isolate \(y\), we start by removing the term \(-3x\) from the left-hand side.
We achieve this by performing the same operation on both sides of the equation.
Adding \(3x\) to both sides results in:
$$2y = 3x + 5$$
Now the term containing \(y\) stands alone on one side of the equation. This process of 'isolating the variable' is crucial as it simplifies the equation, making it easier to solve the next steps.

To isolate variables in general, you'll:

• Undo addition or subtraction first
• Then undo multiplication or division.

This systematic approach ensures you're always moving towards isolating the desired variable from the equation.

linear equations

Linear equations are fundamental in algebra and appear in the form of \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables. The main characteristic of a linear equation is that the variables are not raised to any power other than one.
The given problem, \(-3x + 2y = 5\), is a prime example of a linear equation.
Knowing how to solve linear equations is essential as it forms the basis for understanding more complex algebraic concepts.
Linear equations have important properties, such as:

• They graph as straight lines.
• They can be simplified easily using basic arithmetic operations.
• The solutions of linear equations can often represent real-world phenomena, like calculating slopes in geometry or predicting trends.

Mastery of linear equations allows you to easily transition into more elaborate functions and equations in mathematics.

solving for y

Finally, solving for \(y\) involves getting \(y\) by itself on one side of the equation. This is usually the last step after isolating the \(y\)-term.
From the solution step we had earlier, we are left with:
$$2y = 3x + 5$$
To solve for \(y\), we need to get rid of the coefficient attached to \(y\), which is \(2\) in this case.
We do this by dividing every term in the equation by \(2\):
$$y = \frac{3x + 5}{2}$$
This leaves \(y\) completely by itself on one side of the equation.
We have isolated \(y\) and successfully solved for it!
It's crucial to remember:

• Always perform the same operation on both sides of the equation.
• Check your work by substituting back into the original equation if needed.

Doing so ensures that your solution is correct and helps build confidence in your algebra skills.