Question

Write the equation in slope-intercept form. Then graph the equation. $$ 2 x+3 y-4=x+5 $$

Short answer

The slope-intercept form of the equation is \(y = -\frac{1}{3}x + \frac{5}{3}\). For graphing, start by marking the y-intercept at 5/3 on y-axis, then from there move 1 step right and 1/3 step down (as slope is -1/3) for each subsequent point.

Step-by-step solution

Simplify the equation

Start by combining like terms, subtract \(x\) from both sides of the equation, yielding: \(2x - x + 3y = 5\). This simplifies to \(x + 3y = 5\)

Isolate y

Re-arrange the equation to solve for \(y\). This would involve subtracting \(x\) from both sides and then dividing by 3. The expression will then be \(3y = -x + 5\). Dividing every term by 3 gives \(y = -\frac{1}{3}x + \frac{5}{3}\)

Graph equation

Finally, plot the graph based on the slope-intercept form achieved. The slope here is -1/3, meaning for every one unit of change in x, y decreases by 1/3. The y-intercept is 5/3, indicating the point where the line crosses the y-axis.

Key concepts

Graphing Linear Equations

Graphing linear equations is a fundamental skill in algebra that allows us to visually represent solutions to two-variable equations. A linear equation forms a straight line when graphed on a coordinate plane. The general 'slope-intercept' form of a linear equation is expressed as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) represents the y-intercept.

To graph a linear equation, start by plotting the y-intercept \( b \) on the y-axis. From that point, use the slope \( m \), which is a ratio of rise over run (the change in y over the change in x), to determine the direction of the line. For example, a slope of \( -\frac{1}{3} \) means that for every one unit the x-value increases, the y-value will decrease by \( \frac{1}{3} \) of a unit. After plotting the second point using the slope, draw a line through the two points to extend it across the graph.

For visual clarity and accuracy when graphing a line, you might want to pick another point based on the slope and make sure that this third point aligns with the line through the initial two points. This verification helps in confirming that the slope has been correctly interpreted and applied.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying algebraic expressions and equations to solve for unknown variables. This process often involves several key skills such as combining like terms, using the distributive property, and balancing equations by performing the same operations on both sides.

In the context of converting an equation to slope-intercept form, algebraic manipulation makes the equation easier to graph. For instance, if the original equation is not in the correct form, you would first look to combine any like terms and simplify both sides as required. Following that, rearrange the equation to solve for \( y \). This is done by isolating \( y \) on one side of the equation and placing everything else on the opposite side.

Understanding algebraic manipulation is crucial not just for graphing, but for mastering algebra as a whole. It allows you to solve equations, simplify expressions, and ultimately, to understand the structural relationships between numbers and variables.

Isolating Variables

Isolating variables is one of the key techniques in algebra used for solving equations. This technique involves moving terms containing the variable you are solving for to one side of the equation and all other terms to the opposite side. This process often requires adding, subtracting, multiplying, or dividing both sides of the equation by the same number or expression—ensuring that the equation remains balanced.

To isolate a variable in a linear equation, such as in the example exercise, you would subtract or add terms to get the variable on one side and then use division or multiplication to solve for the variable alone. For example, after simplifying and gathering like terms, if the equation is \( x + 3y = 5 \), you would first subtract \( x \) from both sides to get \( 3y = -x + 5 \). Then, divide every term by 3 to finally isolate \( y \), resulting in \( y = -\frac{1}{3}x + \frac{5}{3} \).

Isolating the variable not only helps in solving equations but also is essential when rearranging equations into a form that allows for easier interpretation, such as the slope-intercept form for graphing purposes. Mastery of this concept is pivotal to success in algebra and beyond.