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Chapter 5: Problem 2

a. Write each linear equation in slope-intercept form. b. From the slope-intercept form, determine the number of solutions to the system. c. Solve the system. $$ \begin{aligned} &2 y-7=-4 x\\\ &6 x+3 y=10 \end{aligned} $$

Short Answer

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Step by step solution

01

Convert First Equation to Slope-Intercept Form

Start with the equation: \[ 2y - 7 = -4x \]Add 7 to both sides: \[ 2y = -4x + 7 \]Divide both sides by 2: \[ y = -2x + \frac{7}{2} \]So, the first equation in slope-intercept form is: \[ y = -2x + \frac{7}{2} \]
02

Convert Second Equation to Slope-Intercept Form

Start with the equation: \[ 6x + 3y = 10 \]Subtract 6x from both sides: \[ 3y = -6x + 10 \]Divide both sides by 3: \[ y = -2x + \frac{10}{3} \]So, the second equation in slope-intercept form is: \[ y = -2x + \frac{10}{3} \]
03

Determine Number of Solutions

Notice both equations are in the form: \[ y = mx + b \]Both have the same slope of -2 but different y-intercepts (\( \frac{7}{2} \) and \( \frac{10}{3} \)). Therefore, the lines are parallel and do not intersect. Hence, there are no solutions to the system.
04

Conclusion

Since the lines do not intersect, the system of equations has no solutions.

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Key Concepts

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

slope-intercept form
The slope-intercept form of a linear equation provides a clear way of expressing straight lines. This form is written as \( y = mx + b \). Here, \(m\) represents the slope, and \(b\) is the y-intercept. The slope indicates how steep the line is, while the y-intercept tells us where the line crosses the \(y\)-axis.

To convert an equation into slope-intercept form, you need to isolate \(y\) on one side of the equation. For instance, consider the equation: \( 2y - 7 = -4x \). Add 7 to both sides: \( 2y = -4x + 7 \). Next, divide both sides by 2: \( y = -2x + \frac{7}{2} \). Now, it's in the correct form \( y = mx + b \) for easy graphing and analysis.
linear equations
Linear equations represent straight lines when graphed. They are typically written in various forms such as slope-intercept (\( y = mx + b \)), standard form (\( Ax + By = C \)), or point-slope form (\( y - y_1 = m (x - x_1) \)).

In linear equations, the highest power of the variable is always 1. For instance, the equation \( 6x + 3y = 10 \) is a linear equation. These equations are fundamental in algebra because they simplify the process of finding intersections, trends, and relationships between variables.
parallel lines
Parallel lines have identical slopes but different y-intercepts. This means they will never cross each other, no matter how far they are extended. In the context of the slope-intercept form \( y = mx + b \), if two lines share the same slope (\(m\)) but different y-intercepts (\(b\)), they are parallel.

For example, let's examine the equations \( y = -2x + \frac{7}{2} \) and \( y = -2x + \frac{10}{3} \). Both have the same slope of \( -2 \) but different y-intercepts (\( \frac{7}{2} \) and \( \frac{10}{3} \)). Therefore, these lines are parallel and, importantly, they never intersect.
no solution
When a system of linear equations results in parallel lines, it signifies that there is no solution. This is because parallel lines never meet, meaning they do not share any common point. As seen in the solved example, both lines had the same slope (\( -2 \)) but different y-intercepts, making them parallel.

To determine if a system of equations has no solution, convert each equation to slope-intercept form and compare their slopes and y-intercepts. If the slopes are identical but y-intercepts differ, then the system has no solution. Understanding this concept is crucial for analyzing more complex algebraic problems and scenarios involving linear equations.

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