Problem 36
Question
Write an equation in slope-intercept form of the line satisfying the given conditions. The line passes through \((2,6)\) and has the same \(y\) -intercept as the line whose equation is \(x-3 y=18.\)
Step-by-Step Solution
Verified Answer
The equation of the line is \(y = 6x - 6\).
1Step 1: Finding the y-intercept of given line
Rewrite the given equation \(x-3y = 18\) in slope-intercept form to find the y-intercept. Solving for \(y\), we have \(3y = x - 18\), hence \(y = \frac{1}{3}x - 6\). The y-intercept is -6.
2Step 2: Calculating the slope of the new line
We have the point \((2,6)\) and the y-intercept \((0,-6)\). So we can find the slope (m) using the formula: \(m=\frac{y2 - y1}{x2 - x1}\). Substituting in the given points, we get \(m=\frac{6 - (-6)}{2 - 0} = 6\).
3Step 3: Write the equation of the line in slope-intercept form
Given that the slope (m) is 6 and the y-intercept (c) is -6, we substitute these values into the slope-intercept form \(y = mx + c\) to get \(y = 6x - 6\). This is the equation of the line.
Key Concepts
Writing Linear EquationsY-interceptSlope CalculationAlgebraic Manipulation
Writing Linear Equations
Understanding how to write linear equations is a crucial skill in algebra. A linear equation forms a straight line when graphed, and the most common way to write it is in the slope-intercept form, which is expressed as
\( y = mx + b \), where
m represents the slope of the line, and b is the y-intercept. The y-intercept is the point where the line crosses the y-axis, and the slope indicates the steepness of the line.
When you're given a point through which the line passes and another equation to refer to, as in the original exercise, the task is to find both values, m and b, to completely determine the desired line's equation. Once you have both the slope and the y-intercept, the process of writing the equation is straightforward. Substitute these values into the slope-intercept form, and you've got the required linear equation.
\( y = mx + b \), where
m represents the slope of the line, and b is the y-intercept. The y-intercept is the point where the line crosses the y-axis, and the slope indicates the steepness of the line.
When you're given a point through which the line passes and another equation to refer to, as in the original exercise, the task is to find both values, m and b, to completely determine the desired line's equation. Once you have both the slope and the y-intercept, the process of writing the equation is straightforward. Substitute these values into the slope-intercept form, and you've got the required linear equation.
Y-intercept
The y-intercept is a fundamental concept when working with linear equations. It's the point where the line crosses the y-axis, and in the equation
\( y = mx + b \), it's represented by b. In our exercise, the y-intercept is determined by manipulating the given secondary equation,
\( x - 3y = 18 \), into slope-intercept form, resulting in
\( y = \frac{1}{3}x - 6 \).The number -6 in this rearranged form is our y-intercept for the line we're seeking. This value is crucial because it doesn't change regardless of where on the line you're looking, making it a reliable starting point for constructing your line equation or for graphing purposes.
\( y = mx + b \), it's represented by b. In our exercise, the y-intercept is determined by manipulating the given secondary equation,
\( x - 3y = 18 \), into slope-intercept form, resulting in
\( y = \frac{1}{3}x - 6 \).The number -6 in this rearranged form is our y-intercept for the line we're seeking. This value is crucial because it doesn't change regardless of where on the line you're looking, making it a reliable starting point for constructing your line equation or for graphing purposes.
Slope Calculation
The slope calculation is an essential step in forming the equation of a line. The slope, typically represented as m, shows how much the y-value of a point on a line changes for a one-unit increase in the x-value. Mathematically, it is computed using the formula
\( m = \frac{y_2 - y_1}{x_2 - x_1} \).In the exercise, we use the slope calculation to find how steep our line will be. We already have one point,
\( (2,6) \), and the y-intercept at
\( (0,-6) \), giving us a second point. Plugging these points into the formula gives us a slope of
\( m = \frac{6 - (-6)}{2 - 0} = 6 \). This value indicates that for every one unit you move to the right along the x-axis, the y-value increases by six units—a pretty steep line!
\( m = \frac{y_2 - y_1}{x_2 - x_1} \).In the exercise, we use the slope calculation to find how steep our line will be. We already have one point,
\( (2,6) \), and the y-intercept at
\( (0,-6) \), giving us a second point. Plugging these points into the formula gives us a slope of
\( m = \frac{6 - (-6)}{2 - 0} = 6 \). This value indicates that for every one unit you move to the right along the x-axis, the y-value increases by six units—a pretty steep line!
Algebraic Manipulation
The term algebraic manipulation encompasses a range of techniques used to rearrange and solve equations. In our context, we apply these techniques to rewrite the given equation in a form that helps us identify the slope and y-intercept.
The given equation was
\( x - 3y = 18 \), which we manipulate to isolate y by dividing all terms by -3. The result is
\( y = \frac{1}{3}x - 6 \), an equation where it's easy to spot the slope and y-intercept. Understanding algebraic manipulation is crucial for solving for these values, which are then used to write the final linear equation in slope-intercept form. It’s important to feel comfortable with operations such as addition, subtraction, multiplication, division, and the use of inverse operations to solve for variables to succeed in algebra.
The given equation was
\( x - 3y = 18 \), which we manipulate to isolate y by dividing all terms by -3. The result is
\( y = \frac{1}{3}x - 6 \), an equation where it's easy to spot the slope and y-intercept. Understanding algebraic manipulation is crucial for solving for these values, which are then used to write the final linear equation in slope-intercept form. It’s important to feel comfortable with operations such as addition, subtraction, multiplication, division, and the use of inverse operations to solve for variables to succeed in algebra.
Other exercises in this chapter
Problem 36
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