Problem 14
Question
Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((1,-1)\) and \((4,5)\)
Step-by-Step Solution
Verified Answer
The equation in standard form is \( 2x - y = 3 \).
1Step 1: Find the Slope of the Line
To determine the slope of the line, we need to use the formula for the slope between two points: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Use the points \((1, -1)\) and \((4, 5)\). Thus, \( m = \frac{5 - (-1)}{4 - 1} = \frac{6}{3} = 2 \).
2Step 2: Use the Point-Slope Form
With the slope \( m = 2 \) and one of the points \((1, -1)\), use the point-slope form of the equation: \( y - y_1 = m(x - x_1) \). Substituting the values gives: \( y - (-1) = 2(x - 1) \). Simplify this equation to get: \( y + 1 = 2x - 2 \).
3Step 3: Simplify to Slope-Intercept Form
Continue simplifying the equation from the point-slope form to slope-intercept form \( y = mx + b \). Simplify \( y + 1 = 2x - 2 \) to \( y = 2x - 2 - 1 \). Therefore, \( y = 2x - 3 \) is the slope-intercept form.
4Step 4: Convert to Standard Form
Standard form is \( Ax + By = C \). Start with the equation \( y = 2x - 3 \), then rearrange to get \( 2x - y = 3 \). Ensure values of \( A, B, \) and \( C \) are integers. Thus, the standard form of the equation is \( 2x - y = 3 \).
Key Concepts
Slope CalculationPoint-Slope FormSlope-Intercept FormStandard Form
Slope Calculation
The slope of a line is a measure of how steep the line is. It's represented by the letter \( m \) in equations. To find the slope between two points on a line, you use the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula calculates the "rise" over the "run," or the change in the vertical direction over the change in the horizontal direction.
- **Calculate the difference in the y-values**: This tells how much the line goes up or down (the rise).
- **Calculate the difference in the x-values**: This indicates how much the line moves from left to right (the run).
For the points \((1, -1)\) and \((4, 5)\), you find the differences as follows:
- **Calculate the difference in the y-values**: This tells how much the line goes up or down (the rise).
- **Calculate the difference in the x-values**: This indicates how much the line moves from left to right (the run).
For the points \((1, -1)\) and \((4, 5)\), you find the differences as follows:
- The difference in y-values is \(5 - (-1) = 6\).
- The difference in x-values is \(4 - 1 = 3\).
- So, the slope \( m \) is \( \frac{6}{3} = 2 \).
Point-Slope Form
The point-slope form of a line equation provides a way to write the equation of a line when you know the slope and at least one point on the line. It is defined as \( y - y_1 = m(x - x_1) \).
- **\( m \)** is the slope of the line.- **\((x_1, y_1)\)** are the coordinates of the given point.
This form is very useful because it directly uses a point on the line and the slope. Let's apply this formula with \( m = 2 \) and the point \((1, -1)\):
- Substitute to get: \( y - (-1) = 2(x - 1) \).
- Simplify to: \( y + 1 = 2x - 2 \).
By simplifying, you can easily convert it into other forms of line equations, like the slope-intercept form.
- **\( m \)** is the slope of the line.- **\((x_1, y_1)\)** are the coordinates of the given point.
This form is very useful because it directly uses a point on the line and the slope. Let's apply this formula with \( m = 2 \) and the point \((1, -1)\):
- Substitute to get: \( y - (-1) = 2(x - 1) \).
- Simplify to: \( y + 1 = 2x - 2 \).
By simplifying, you can easily convert it into other forms of line equations, like the slope-intercept form.
Slope-Intercept Form
The slope-intercept form is a convenient way to express a line equation because it clearly shows the slope and the y-intercept. It's written as \( y = mx + b \), where:
- **\( m \)** is the slope.- **\( b \)** is the y-intercept (where the line crosses the y-axis).
After simplifying the point-slope equation \( y + 1 = 2x - 2 \):
- **\( m \)** is the slope.- **\( b \)** is the y-intercept (where the line crosses the y-axis).
After simplifying the point-slope equation \( y + 1 = 2x - 2 \):
- Subtract 1 from both sides to get \( y = 2x - 3 \).
Standard Form
The standard form of a line's equation is \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) should be non-negative.
This form is useful in various mathematical analyses, such as calculating intersections or transformations. To convert from slope-intercept to standard form:
- Start with \( y = 2x - 3 \).- Rearrange terms to get: \( 2x - y = 3 \).
In this form, notice how all terms are on one side of the equation, equal to a constant. This makes it easier to compare and manipulate equations in algebraic contexts.
Standard form provides a straightforward method for finding intercepts and working with systems of equations.
This form is useful in various mathematical analyses, such as calculating intersections or transformations. To convert from slope-intercept to standard form:
- Start with \( y = 2x - 3 \).- Rearrange terms to get: \( 2x - y = 3 \).
In this form, notice how all terms are on one side of the equation, equal to a constant. This makes it easier to compare and manipulate equations in algebraic contexts.
Standard form provides a straightforward method for finding intercepts and working with systems of equations.
Other exercises in this chapter
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