Problem 69 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

Without graphing the equations, we can observe that they share the same slope (-3), but have different y-intercepts. Thus, the lines are parallel and never intersect. So, the system has no solutions.

Step 1: Convert to Slope-Intercept Form

Convert the two equations in system to slope-intercept form (y = mx + b):For the first equation, '6x + 2y = 3', it becomes 'y = -3x + 3/2'. For the second equation, '3x + y = -2', it becomes 'y = -3x - 2'.

Step 2: Graph the Equations

Next, draw the graph for each equation in a plain paper. Use the slope (-3) and the y-intercept (3/2 and -2 respectively) to plot the lines.

Step 3: Identify the Intersections

Look for the points where the two lines intersect: If they intersect in exactly one point, then there is one solution to the system. If they overlay each other (are the same line), then there are infinitely many solutions.If they never intersect (are parallel), then there are no solutions to the system.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial when dealing with linear equations. Simply put, it is a way of writing the equation of a line so that it is easy to graph. This form is expressed as y = mx + b, where m represents the slope of the line, and b is the y-intercept—the point where the line crosses the y-axis.

When converting a standard form equation, like 6x + 2y = 3, into slope-intercept form, the goal is to isolate y. Here's how it's done in a step-by-step fashion:

• Divide every term by the coefficient of y, which is 2 in this case, giving us 3x + y = 1.5.
• Subtract 3x from both sides to get y by itself, resulting in y = -3x + 1.5.

Doing this allows you to quickly identify the slope and y-intercept, which are essential for graphing the equation. The same process applies to any linear equation you want to graph.

Graphing Linear Equations

Graphing linear equations involves plotting lines on a coordinate system based on their equations. This process can be significantly simplified by using the slope-intercept form. Once you have an equation like y = -3x + 1.5, you start by marking the y-intercept on the vertical axis. In our example, this would be the point (0, 1.5).

Next, use the slope, which indicates the rise over the run, to determine the next point on the line. A slope of -3 means that for every 3 units down (negative rise), you move 1 unit to the right (positive run). With these two points, you can draw a line extending in both directions. Repeat this process for the second equation, and observe the resulting graphs.

Creating an Accurate Graph

If drawing by hand, it is important to use graph paper and a ruler for accuracy. For students struggling to plot points accurately, using online graphing tools or calculators can also be invaluable. With practice, graphing becomes an intuitive part of solving equations!

Solutions to Systems of Equations

A 'solution' to a system of equations refers to a set of values for the variables that make all equations true simultaneously. When graphing, the solution is visually represented by the point or points where the lines intersect. There are three possibilities:

• If the lines intersect at exactly one point, the system has one unique solution. Both equations are satisfied only at that point.
• If the lines overlap completely, they are essentially the same line, which means there are infinitely many solutions.
• If the lines are parallel and never intersect, the system has no solution.

In the provided exercise, both equations, after being converted to slope-intercept form, have the same slope but different y-intercepts. This means they are parallel lines, leading to the conclusion that the system has no solutions. For students, visualizing the graph can aid in understanding why the lack of intersection points corresponds to having no solutions.