Problem 64 - Algebra Form and Function

Question

Which line has the greater (a) Slope? (b) \(y\) -intercept? $$ 3 y=5 x-2, \quad y=2 x+1 $$

Step-by-Step Solution

Answer

Answer: (a) Line 1 has a greater slope. (b) Line 2 has a greater y-intercept.

Step 1: Convert the given lines into slope-intercept form

First, let's rewrite both lines in slope-intercept form: Line 1: \(3y = 5x - 2\) To isolate y, we can divide the whole equation by 3: $$ y = \frac{5}{3}x -\frac{2}{3} $$ Line 2: \(y = 2x + 1\) (already in slope-intercept form) Now we have both lines in slope-intercept form: Line 1: \(y = \frac{5}{3}x -\frac{2}{3}\) Line 2: \(y = 2x + 1\)

Step 2: Compare the slopes

Now that we have both equations in slope-intercept form, we can easily compare their slopes by looking at the coefficients of x. For Line 1: The slope (\(m_1\)) is \(\frac{5}{3}\). For Line 2: The slope (\(m_2\)) is 2. Now, let's compare the slopes: $$ \frac{5}{3} > 2 \Rightarrow m_{1}>m_{2} $$ Since \(\frac{5}{3} > 2\), Line 1 has a greater slope.

Step 3: Compare the y-intercepts

Next, we'll compare the y-intercepts, which are the constants in the slope-intercept form of the equations. For Line 1: The y-intercept (\(b_1\)) is \(-\frac{2}{3}\). For Line 2: The y-intercept (\(b_2\)) is 1. Now, let's compare the y-intercepts: $$ -\frac{2}{3} < 1 \Rightarrow b_{1}

Key Concepts

Slope Comparison

When dealing with linear equations, one of the key aspects to understand is the slope. The slope of a line tells us how steep the line is. It is the number that appears in front of the variable \(x\) when the equation is in the slope-intercept form, \(y = mx + b\). Here, \(m\) is the slope, and it indicates the rate of change of \(y\) with respect to \(x\).
A larger slope means the line is steeper. For instance, in our scenario, we have:

• For Line 1: Slope \(m_1 = \frac{5}{3}\).
• For Line 2: Slope \(m_2 = 2\).

To compare them, we look at \(\frac{5}{3}\) and \(2\). Even though \(\frac{5}{3}\) may look larger because of the fraction, it is actually less than \(2\) when converted to a decimal, as \(\frac{5}{3} \approx 1.67\).
From this, we conclude that Line 1 is less steep than Line 2, meaning Line 2 actually has the greater slope.

Y-Intercept Comparison

The y-intercept is another critical characteristic of a linear equation in its slope-intercept form \(y = mx + b\). The y-intercept \(b\) is the point where the line crosses the y-axis, essentially meaning that it is the value of \(y\) when \(x\) is zero.

• For Line 1: Y-intercept \(b_1 = -\frac{2}{3}\).
• For Line 2: Y-intercept \(b_2 = 1\).

We compare \(-\frac{2}{3}\) and \(1\). Here, \(-\frac{2}{3}\) is a negative number, while \(1\) is positive. This clearly indicates that the y-intercept of Line 2 is greater than that of Line 1.
In practical terms, this means Line 2 starts above the x-axis, whereas Line 1 starts below it.

Linear Equations

Linear equations are foundational in mathematics, representing straight lines on a graph. They are typically written in slope-intercept form: \(y = mx + b\). Here, \(m\) represents the slope, and \(b\) represents the y-intercept. This form makes it easy to quickly identify and compare key features of the line.
When writing equations for lines, converting them to slope-intercept form helps in easily determining their characteristics. For example, converting \(3y = 5x - 2\) to \(y = \frac{5}{3}x - \frac{2}{3}\) involves isolating \(y\), revealing the slope and y-intercept.

Linear equations like these are used in various applications such as predicting trends, analyzing rates of change, and solving practical problems across numerous fields like physics, economics, and more. Understanding them provides a basis for more advanced mathematics and applications in everyday life.

• They graph as straight lines.
• They have constant slopes.
• They intersect the y-axis at a specific point, known as the y-intercept.

Mastering linear equations is crucial for mathematical literacy and problem-solving skills.

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Problem 63

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Problem 65

Other exercises in this chapter

Problem 62

Write an expression in \(x\) representing the result of the given operations on \(x\). Is the expression linear in \(x ?\) Add \(x,\) multiply by \(5,\) subtrac

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Problem 63

Which line has the greater (a) Slope? (b) \(y\) -intercept? $$ 2 x=4 y+3, \quad y=-x-2 $$

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Problem 65

Which line has the greater (a) Slope? (b) \(y\) -intercept? $$ y+2=3(x-1), \quad y=6-50 x $$

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Problem 66

Which line has the greater (a) Slope? (b) \(y\) -intercept? $$ y-3=-4(x+2), \quad-2 x+5 y=-3 $$

View solution

Key Concepts

Slope Comparison

When dealing with linear equations, one of the key aspects to understand is the slope. The slope of a line tells us how steep the line is. It is the number that appears in front of the variable \(x\) when the equation is in the slope-intercept form, \(y = mx + b\). Here, \(m\) is the slope, and it indicates the rate of change of \(y\) with respect to \(x\).
A larger slope means the line is steeper. For instance, in our scenario, we have:

• For Line 1: Slope \(m_1 = \frac{5}{3}\).
• For Line 2: Slope \(m_2 = 2\).

To compare them, we look at \(\frac{5}{3}\) and \(2\). Even though \(\frac{5}{3}\) may look larger because of the fraction, it is actually less than \(2\) when converted to a decimal, as \(\frac{5}{3} \approx 1.67\).
From this, we conclude that Line 1 is less steep than Line 2, meaning Line 2 actually has the greater slope.

Y-Intercept Comparison

The y-intercept is another critical characteristic of a linear equation in its slope-intercept form \(y = mx + b\). The y-intercept \(b\) is the point where the line crosses the y-axis, essentially meaning that it is the value of \(y\) when \(x\) is zero.

• For Line 1: Y-intercept \(b_1 = -\frac{2}{3}\).
• For Line 2: Y-intercept \(b_2 = 1\).

We compare \(-\frac{2}{3}\) and \(1\). Here, \(-\frac{2}{3}\) is a negative number, while \(1\) is positive. This clearly indicates that the y-intercept of Line 2 is greater than that of Line 1.
In practical terms, this means Line 2 starts above the x-axis, whereas Line 1 starts below it.

Linear Equations

Linear equations are foundational in mathematics, representing straight lines on a graph. They are typically written in slope-intercept form: \(y = mx + b\). Here, \(m\) represents the slope, and \(b\) represents the y-intercept. This form makes it easy to quickly identify and compare key features of the line.
When writing equations for lines, converting them to slope-intercept form helps in easily determining their characteristics. For example, converting \(3y = 5x - 2\) to \(y = \frac{5}{3}x - \frac{2}{3}\) involves isolating \(y\), revealing the slope and y-intercept.

Linear equations like these are used in various applications such as predicting trends, analyzing rates of change, and solving practical problems across numerous fields like physics, economics, and more. Understanding them provides a basis for more advanced mathematics and applications in everyday life.

• They graph as straight lines.
• They have constant slopes.
• They intersect the y-axis at a specific point, known as the y-intercept.

Mastering linear equations is crucial for mathematical literacy and problem-solving skills.