Question

Use the graphing method to solve the linear system and tell how many solutions the system has. $$ \begin{aligned}&6 x-2 y=4\\\&-4 x+2 y=-\frac{8}{3}\end{aligned} $$

Short answer

This system of linear equations has no solution, since the lines they represent are parallel.

Step-by-step solution

Put into slope-intercept form

Start by transforming both equations into the slope-intercept form. For the first equation, 6x - 2y = 4, isolate y by subtracting 6x from both sides and then dividing by -2 to get \(y = 3x -2\). For the second equation, -4x + 2y = -8/3, add 4x to both sides and divide by 2 to get \(y = 2x + 4/3\)

Graph

Next you graph both equations on the same coordinate grid. The first line has a slope of 3 and a y-intercept of -2. This means it starts at point (0,-2) and goes up 3 units for every unit it goes to the right. The second line has a slope of 2 and a y-intercept of 4/3. This means it starts at point (0,4/3) and goes up 2 units for every unit it goes to the right.

Analyze the graph

If you look at your graph, you'll see that the lines are parallel to each other. This means they will never intersect.

Key concepts

Slope-Intercept Form

When working with linear equations, one of the most convenient forms to express them is the slope-intercept form, which is written as \( y = mx + b \). Here, \( m \) represents the slope of the line, which tells you how steep the line is, and \( b \) represents the y-intercept, the point where the line crosses the y-axis.

Converting to slope-intercept form simplifies the process of graphing because it directly provides the starting point (\( b \)) and the direction and steepness of the line (\( m \)). For example, in our exercise, the first equation is converted to \( y = 3x - 2 \), indicating a y-intercept at \( -2 \) and a slope of \( 3 \), which means for every one unit you move to the right along the x-axis, you move up three units along the y-axis.

Graphing Linear Equations

Graphing linear equations is an effective visual method to understand the relationship between two variables. It involves plotting points on a graph where each point represents a solution to the equation.

To graph equations in slope-intercept form, start by plotting the y-intercept on the y-axis. Then, use the slope to find other points on the line. For the slope \( \frac{rise}{run} \), 'rise' refers to the vertical change, and 'run' refers to the horizontal change.

In our problem, for the line with equation \( y = 3x - 2 \), you'd plot the y-intercept at (0, -2) and then from this point, go up 3 units and to the right 1 unit to find another point on the line. Repeat this to find as many points as needed, then draw a line through these points. Repeat the process for the second equation to graph the second line.

Parallel Lines

Parallel lines are lines in a plane that never intersect; they remain the same distance apart over their entire length. In terms of slope, parallel lines have identical slopes, meaning the change in y over the change in x is the same for both lines.

In the given exercise, after graphing both lines, you can observe they do not intersect. This is because they are parallel, indicated by their slopes. The first line has a slope of \( 3 \), and the second line has a slope of \( 2 \). If two lines have different slopes, they will eventually meet at some point, but if the slopes are the same, they will never come together, which in this case indicates that the system of equations has no solution.