Problem 6 - Algebra 1: Concepts and Skills

Question

Determine whether the equation is in slope-intercept form. $$ x+23 y=-15 $$

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = -\frac{1}{23}x - \frac{15}{23}\).

Step 1: Identify the given equation

The given equation is \(x+23y=-15\).

Step 2: Rearrange the equation into slope-intercept form

To get the equation into slope-intercept form, isolate the term involving \(y\) on one side of the equation. This involves subtracting \(x\) from both sides of the equation, which results in \(23y = -x - 15\).

Step 3: Simplify the equation

Finally, divide every term by 23 to solve for \(y\). This gives the equation in slope-intercept form: \(y = -\frac{1}{23}x - \frac{15}{23}\).

Key Concepts

Linear Equations

Linear equations are foundational in algebra and represent straight lines on a graph. They consist of variables and constants, and the highest power of the variable is never greater than one. In its simplest form, a linear equation in two variables, like 'x' and 'y', can be expressed as \(y = mx + b\), where 'm' represents the slope of the line and 'b' represents the y-intercept, which is the point where the line crosses the y-axis. The slope 'm' defines the steepness and the direction of the line: a positive slope means the line goes upwards from left to right, while a negative slope means it goes downwards. When the equation does not look like the standard form, algebraic manipulation is often necessary to rearrange it.

Algebraic Manipulation

Algebraic manipulation involves using mathematical operations to reshape equations without changing their solutions or the relationships between variables. This skill is paramount in solving and graphing linear equations. Reaching the slope-intercept form from any linear equation requires such manipulations, which can include adding, subtracting, multiplying, or dividing both sides of the equation by the same number. Operations also have to 'undo' each other properly; for example, if there is a term subtracting from 'y', you add it to both sides to keep the equation balanced.

• Addition/Subtraction: Used to move terms from one side of the equation to the other.
• Multiplication/Division: Used to isolate the variable and solve for it.

For instance, in the given exercise, subtracting 'x' from both sides moved it to the right, thus isolating the terms with 'y' on the left. Then, division by 23 was necessary to solve for 'y'.

Equation Rearrangement

Equation rearrangement is the process of shaping an equation into a more useful or recognizable form. For linear equations, one common goal is to rearrange them into the slope-intercept form. This form is highly useful for graphing the equation swiftly and understanding its characteristics with ease. The slope-intercept form makes it instantly clear what the slope and y-intercept of the line are.
In the provided exercise, rearrangement required subtracting 'x' and then dividing every term by 23 to isolate 'y'. This alters the form of the equation but retains the same graph and solutions. The final form, \(y = -\frac{1}{23}x - \frac{15}{23}\), is in the desired slope-intercept format and clearly shows the slope as \( -\frac{1}{23} \) and the y-intercept as \( -\frac{15}{23} \), thus fully preparing us to graph the line or understand its behavior in relation to the coordinate system.