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Chapter 3: Problem 78

Graph equation. Solve for \(y\) first, when necessary. \(6 y=-4 x\)

Short Answer

Expert verified
The solution for \(y\) is \(y = \frac{-2}{3}x\). The slope is \(-\frac{2}{3}\) and y-intercept is 0.

Step by step solution

01

Isolate y

To solve for \(y\), we want to get \(y\) by itself on one side of the equation. We start with the equation \(6y = -4x\). Divide both sides by 6 to isolate \(y\):\[ y = \frac{-4x}{6} \]
02

Simplify the Equation

Now we simplify the fraction on the right side of the equation. Divide both the numerator and the denominator by 2:\[ y = \frac{-2x}{3} \]This is the simplified form of the equation where \(y\) is isolated.
03

Determine the Slope and Intercept

The equation is now in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. From \(y = \frac{-2}{3}x\), we identify the slope \(m = -\frac{2}{3}\) and the y-intercept \(b = 0\). This means the line passes through the origin.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing
Graphing is a method of visually representing equations, inequalities, or data sets on a coordinate plane to show the relationships between variables. To graph a linear equation like the one we've solved, plot points on the coordinate plane to represent the solutions. These points align in a straight line, illustrating the linear nature of the equation.

When graphing, it's essential to consider particular points:
  • Intercepts: Points where the line crosses the axes. In our equation, since the y-intercept is 0, the line passes through the origin.
  • Slope: The steepness of the line, showing the rate of change between the variables.
By identifying these features, you can accurately plot the graph of the equation, helping visualize solutions and relations between variables.
Slope-Intercept Form
The slope-intercept form of a linear equation is \[ y = mx + b \] where:
  • m is the slope, representing the ratio of the vertical change to horizontal change between any two points on the line.
  • b is the y-intercept, which marks the point where the line crosses the y-axis.
Using our example, \[ y = -\frac{2}{3}x \], the slope \( m \) is \(-\frac{2}{3}\), telling us the line moves downward two units for every three units it moves right.

The absence of \( b \) (i.e., it is zero) means the line crosses the y-axis at the origin. Understanding this form lets you quickly determine how the line behaves and where it鈥檚 positioned on the graph.
Solving for Variables
Solving for variables is a key skill in algebra, allowing us to find unknown values by rearranging equations. For the equation \( 6y = -4x \), we solved for \( y \) by isolating it on one side.

To isolate \( y \), divide both sides by 6, resulting in:\[ y = \frac{-4x}{6} \]. This simplifies to\[ y = \frac{-2}{3}x \]. Each step involves careful manipulation, ensuring no change in meaning. This process is crucial for expressing relations in understandable terms and can be applied to equations of various complexities, forming the foundation of solving algebraic equations.

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