Problem 29 - Algebra 2

Question

Graph and solve each system. Where necessary, estimate the solution. $$ \left\\{\begin{array}{l}{3 x+y=3} \\ {2 x-y=7}\end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system is approximately the point of intersection of the two lines when graphed. Estimate the coordinates if the graph isn't precise.

Step 1: Rewrite the equations in slope-intercept form

Rewrite both equations of the system in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. First equation: y = -3x + 3 Second equation: y = 2x - 7

Step 2: Graph the first equation

Plot the y-intercept of the first equation (0,3) on the coordinate plane. Then, use the slope -3 to find another point. From (0,3), move 3 units down and 1 unit to the right since the slope is negative. Plot the second point and draw the line through these two points.

Step 3: Graph the second equation

For the second equation, plot the y-intercept (0,-7) on the same coordinate plane. Use the slope of 2 to find a second point. Move 2 units up and 1 unit to the right from (0,-7). Plot this point and draw the line through the two points.

Step 4: Find the point of intersection

The solution to the system of equations is where the two lines intersect on the graph. Look for the point where both lines cross. This point is the pair (x, y) that satisfies both equations. If the graph is not precise, try to estimate the coordinates of the intersection to the nearest tenth.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is foundational when solving systems of equations graphically. In its essence, the slope-intercept form of a linear equation is expressed as \( y = mx + b \). This compact representation provides two vital pieces of information about a line: its slope (\( m \)) and its y-intercept (\( b \)).

The slope, \( m \), indicates the steepness or incline of the line and directs you on how to move from one point on the line to another. For instance, a slope of 2 means for every unit you move to the right, you'll ascend vertically by 2 units, creating an 'uphill' pattern. Conversely, a negative slope implies a 'downhill' trajectory.

The y-intercept, \( b \), reveals where the line crosses the y-axis. This point is vital for plotting the starting coordinate of the line. Once you have the y-intercept, you use the slope to find another point on the line and then draw the line through both points.

Graphing Linear Equations

Graphing linear equations is a visual method of understanding the relationships between variables. When graphing, we usually begin by plotting the y-intercept on the Cartesian plane. This serves as our starting point and corresponds to where the line crosses the y-axis at \( x = 0 \).

Next, we apply the slope to locate another point on the line. If the slope is positive, we move up and to the right from the y-intercept; if it's negative, we move down and to the right. Repeating this movement according to the magnitude of the slope creates a series of points that lie on the same line. By connecting these points, we visually depict the linear equation.

Here's a quick tip: If the slope is a fraction, like \(\frac{2}{3}\), move up 2 units and right 3 units. This helps with accuracy when plotting points. And remember, vertical lines are special; they have an undefined slope and are represented by an equation like \( x = a \), where \( a \) is a constant.

Point of Intersection

The point of intersection is the precise point where two lines on a graph cross each other. It represents the set of coordinates that simultaneously satisfies both equations in a system. When graphing by hand, pinpointing the exact point of intersection can sometimes be challenging. Estimating to the nearest tenth can be effective when an exact solution isn't easily determined from the graph.

To solve a system of equations graphically, we plot both equations on the same set of axes. The point where the two lines intersect represents the solution to the system. If the lines intersect, we have one solution—this is known as a consistent and independent system. If the lines are parallel and never intersect, this indicates an inconsistent system with no solution. And lastly, if the lines coincide, every point of intersection is a solution, signifying a dependent system. Recognizing the point of intersection is key to understanding the relationship between the two lines and unravelling the system of equations.