Problem 64 - Algebra A Combined Function

Question

Solve each equation for y. See Section 2.5. $$ 5 x+2 y=7 $$

Step-by-Step Solution

Answer

y = -\frac{5}{2}x + \frac{7}{2}

Step 1: Isolate the Term with y

We start by isolating the term with \(y\) in the given equation. Begin by subtracting \(5x\) from both sides of the equation: \[ 5x + 2y = 7 \] Subtract \(5x\) from each side: \[ 2y = -5x + 7 \]

Step 2: Solve for y

Now that \(2y\) is isolated, our next step is to solve for \(y\). We can do this by dividing each term in the equation by \(2\): \[ y = -\frac{5}{2}x + \frac{7}{2} \] The equation \(y = -\frac{5}{2}x + \frac{7}{2}\) is in the form \(y = mx + b\), representing a linear equation.

Key Concepts

Isolating Variables

In algebra, isolating variables is an essential step when solving equations. The goal is to get the variable you're solving for, in this case, \(y\), by itself on one side of the equation. To isolate a variable, you must perform operations that reverse the operations being applied to it.
For example, if a term is being added, you'll need to subtract it from both sides of the equation.

• In our example, the initial equation is \(5x + 2y = 7\).
• We start by subtracting \(5x\) from both sides to isolate the term that contains \(y\):
• \(2y = -5x + 7\).

By doing this, the equation becomes simpler, and you can proceed to the next step, which is solving for the variable you've isolated. The key is to maintain the balance of the equation by doing the same operation to both sides.

Linear Equation Form

Linear equations describe a straight line when plotted on a coordinate plane. The standard linear equation form is typically written as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants. Understanding this form is crucial because it helps identify the relationship between variables, typically \(x\) and \(y\).

• For our example, the initial equation \(5x + 2y = 7\) is in linear form.
• This equation indicates a linear relationship between \(x\) and \(y\).
• The coefficients \(A = 5, B = 2,\) and the constant \(C = 7\) give us insight into the line's slope and position.

A linear equation in this form is versatile for different algebraic manipulations, including transforming it to other forms such as the slope-intercept form, making it easier to graph and interpret.

Slope-Intercept Form

The slope-intercept form of a linear equation, \(y = mx + b\), is perhaps the most popular format for showing linear relationships. It makes it very easy to graph a line because it explicitly provides the slope \(m\) and the y-intercept \(b\).

• The slope \(m\) tells us the steepness of the line, indicating how much \(y\) changes for a unit change in \(x\).
• The y-intercept \(b\) is the point where the line crosses the y-axis.

In the example \(y = -\frac{5}{2}x + \frac{7}{2}\):

• The slope \(m\) is \(-\frac{5}{2}\), meaning for every one unit increase in \(x\), \(y\) decreases by \(\frac{5}{2}\) units.
• The y-intercept \(b\) is \(\frac{7}{2}\), showing that the line crosses the y-axis at this point.

Graphing this equation would reveal the line's characteristics, making this form universally admired for its simplicity in visualization and analysis.