Problem 24
Question
Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line with slope 1 and \(x\) -intercept ( 2,0 )
Step-by-Step Solution
Verified Answer
The equation in standard form is \( x - y = 2 \).
1Step 1: Identify the slope and x-intercept
The problem states that the line has a slope of 1 and passes through the x-intercept (2, 0). We need this information to write the equation of the line.
2Step 2: Use the Point-Slope Form of a Line
The point-slope form of a linear equation is given by \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. Plug in \( m = 1 \), \( x_1 = 2 \), and \( y_1 = 0 \). This gives us \( y - 0 = 1(x - 2) \) or \( y = x - 2 \).
3Step 3: Convert to Standard Form
Standard form of a line is \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) should be positive. Starting from \( y = x - 2 \), we can rearrange it to \( -x + y = -2 \). Adding \( x \) to both sides, we get \( x - y = 2 \).
4Step 4: Ensure Integers in Standard Form
Ensure that the coefficients are integers and that the leading coefficient \( A \) is positive. Here, the equation \( x - y = 2 \) already satisfies the condition for standard form, with \( A = 1 \), \( B = -1 \), and \( C = 2 \).
Key Concepts
Slope-Intercept FormPoint-Slope FormStandard Form
Slope-Intercept Form
The slope-intercept form is a common way to write the equation of a line. This format is particularly useful because it clearly shows the slope and y-intercept of the line. The general formula is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
When you know the slope of a line and where it crosses the y-axis, you can immediately write the equation in this form. For example, if a line has a slope of 1 and crosses the y-axis at -2, the equation is \( y = 1x - 2 \) or simply \( y = x - 2 \).
This form is particularly useful for quickly graphing a line or understanding how the line behaves with changes in \( x \). The value \( m \) shows how steep the line is, and positive values mean the line ascends to the right, while negative values descend.
When you know the slope of a line and where it crosses the y-axis, you can immediately write the equation in this form. For example, if a line has a slope of 1 and crosses the y-axis at -2, the equation is \( y = 1x - 2 \) or simply \( y = x - 2 \).
This form is particularly useful for quickly graphing a line or understanding how the line behaves with changes in \( x \). The value \( m \) shows how steep the line is, and positive values mean the line ascends to the right, while negative values descend.
Point-Slope Form
Point-slope form is beneficial when you know the slope of a line and one point through which it passes. The formula for point-slope form is \( y - y_1 = m(x - x_1) \). Here, \((x_1, y_1)\) is a point on the line, and \( m \) is the slope.
This form is especially handy when you don't have the y-intercept but know another point instead. For instance, given a slope of 1 and an x-intercept of (2, 0), you could directly use point-slope form: \( y - 0 = 1(x - 2) \). Simplifying this gives you \( y = x - 2 \).
Point-slope form provides insight into how the line changes with x while anchoring it to a known point. It's a flexible form that can be converted to slope-intercept form if needed, as well as directly employed to find other line characteristics.
This form is especially handy when you don't have the y-intercept but know another point instead. For instance, given a slope of 1 and an x-intercept of (2, 0), you could directly use point-slope form: \( y - 0 = 1(x - 2) \). Simplifying this gives you \( y = x - 2 \).
Point-slope form provides insight into how the line changes with x while anchoring it to a known point. It's a flexible form that can be converted to slope-intercept form if needed, as well as directly employed to find other line characteristics.
Standard Form
The standard form of a line equation is another way to express linear equations. The general equation is \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) should be positive.
To convert equations from slope-intercept or point-slope to standard form, rearrange the terms to fit \( Ax + By = C \). From the earlier steps, if we start with \( y = x - 2 \), moving the terms gives \( -x + y = -2 \). Making \( A \) positive by adding \( x \) to both sides results in \( x - y = 2 \), which is in standard form.
The standard form is particularly useful in mathematical operations like systems of equations or when you need integer coefficients. It also gives a uniform representation that can be useful in both algebraic manipulations and analysis of line intersections.
To convert equations from slope-intercept or point-slope to standard form, rearrange the terms to fit \( Ax + By = C \). From the earlier steps, if we start with \( y = x - 2 \), moving the terms gives \( -x + y = -2 \). Making \( A \) positive by adding \( x \) to both sides results in \( x - y = 2 \), which is in standard form.
The standard form is particularly useful in mathematical operations like systems of equations or when you need integer coefficients. It also gives a uniform representation that can be useful in both algebraic manipulations and analysis of line intersections.
Other exercises in this chapter
Problem 24
Explain how the following functions can be obtained from \(y=x^{3}\) by basic transformations: (a) \(y=x^{3}-1\) (b) \(y=-x^{3}-1\) (c) \(y=-3(x-1)^{3}\)
View solution Problem 24
(a) Graph \(f(x)=x, x \geq 0\), and \(g(x)=x^{2}, x \geq 0\), together, in one coordinate system. (b) For which values of \(x\) is \(f(x) \geq g(x)\), and for w
View solution Problem 25
Explain how the following functions can be obtained from \(y=1 / x\) by basic transformations: (a) \(y=1-\frac{1}{x}\) (b) \(y=-\frac{1}{x-1}\) (c) \(y=\frac{x}
View solution Problem 25
(a) Graph \(f(x)=x^{2}\) and \(g(x)=x^{3}\) for \(x \geq 0\), together, in one coordinate system. (b) Show algebraically that \(x \geq x^{2}\) for \(0 \leq x \l
View solution