Question

Rewrite the equation in slope-intercept form. $$3 x-7 y=42$$

Short answer

The equation rewritten in slope-intercept form is \(y = \frac{3}{7}x - 6\).

Step-by-step solution

Isolate y on one side

First, start by isolating \(y\) in the equation by moving all terms containing \(x\) to the other side. This gives you: \(-7y = -3x + 42\).

Solve for y

Next, to solve for \(y\), divide every term in the equation by -7. This gives you: \(y = \frac{3}{7}x - 6\).

Verify Slope-Intercept Form

Finally, ensure that the equation is now in slope-intercept form. The equation \(y = \frac{3}{7}x - 6\) fits the format \(y = mx + b\), with \(m = \frac{3}{7}\) and \(b = -6\).

Key concepts

Solving Linear Equations

When dealing with algebra, you'll often find yourself tackling linear equations, which are equations of the first degree. These involve variables raised to no power higher than one and can usually be graphed as straight lines. The key is to manipulate the equation to isolate the variable of interest; in your case, it's the variable 'y'.

Approach solving these equations with the goal of making 'y' the subject of the formula, which means all other terms should be moved to the opposite side of the equal sign. Remember, whatever operation you apply to one side must also be applied to the other to maintain balance. Divide or multiply to manage coefficients, and add or subtract to deal with constants. This methodical process is the golden rule of algebra and is crucial for reaching the correct solution.

Algebraic Manipulation

Algebra requires you to be savvy with a set of manipulation tools so that you can rearrange equations effectively. In your case, coming from a standard linear equation to the slope-intercept form is a two-step game of algebraic manipulation.

First, you moved the term with 'x' to the other side by adding '3x' to both sides, achieving

Isolation of y

This is followed by addressing the coefficient of 'y'. Since the equation had a '-7' as the coefficient, you divided every term by '-7' to get y by itself, resulting in:

Division for Simplification

It's crucial to master these manipulation skills as they are foundational in algebra and pave the way to solving a vast array of problems.

Graphing Linear Equations

Graphing is a visual means to represent your equations and understand the relationship between variables. The slope-intercept form of a line, given as 'y = mx + b', directly provides you with two vital pieces of graphical information: the slope, 'm', and the y-intercept, 'b'.

In your solved equation,

The Slope

is

The Y-intercept

draws a line on graph paper starting at point (0, b) and uses the slope to determine the steepness and direction. If 'm' is positive, the line inclines upwards; if negative, it declines. The beauty of the slope-intercept form is its direct translation to graphing, making it highly valuable for visual learners and a key aspect of mathematics education.