Problem 29 - Elementary Algebra Within Reach

Question

In Exercises 27-36, solve the system by graphing. $$ \left\\{\begin{array}{r} -4 x+2 y=12 \\ 2 x+y=6 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system of equations is (1, 4).

Step 1: Convert equations into slope-intercept form

Equation 1: -4x + 2y = 12 can be written as y = 2x + 6. Equation 2: 2x + y = 6, can be rewritten as y = -2x + 6.

Step 2: Identify slope and y-intercept

For Equation 1, y = 2x + 6, the slope(m1) is 2 and y-intercept(b1) is 6. For equation 2, y = -2x + 6, the slope(m2) is -2 and y-intercept(b2) is 6.

Step 3: Plot the lines

Plotting the line for each equation using their slopes and y-intercepts on the same set of axes will reveal the solution. The coordinates where the lines intersect is the solution to the system.

Step 4: Find the solution

Locate the intersection point of the two lines which is (1, 4). This is the solution to the system of equations.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most commonly used techniques to represent a straight line. It is written as:
\[ y = mx + b \]
Here, 'm' represents the slope of the line, which indicates the steepness and direction of the line. The 'b' represents the y-intercept, which is the point where the line crosses the y-axis.

To convert a standard form equation (like \(-4x + 2y = 12\)) to slope-intercept form, you need to isolate the 'y' term on one side. For instance:

\[ -4x + 2y = 12 \]
\[ 2y = 4x + 12 \]
\[ y = 2x + 6 \]
This equation now tells us that for each increase of 1 in 'x', 'y' will increase by 2, which is the slope. It also tells us that when 'x' is 0, 'y' will be 6 - this is the point (0, 6) on the y-axis.

Graphing Linear Equations

Graphing linear equations involves plotting the lines on a coordinate grid to visualize the relationship between two variables. The slope-intercept form greatly simplifies this process, as it provides the initial value at which the line crosses the y-axis (the y-intercept), and the slope indicates how the line rises or falls as it moves from left to right.

Using the slope, or 'rise over run' approach, you can plot additional points from the y-intercept. The equation \(y = 2x + 6\) has a slope (m) of 2, meaning for every unit we move to the right on the 'x' axis, we move up two units on the 'y' axis. Similarly, a negative slope, as in \(y = -2x + 6\), would indicate a downward trajectory to the right.

After finding a few points with the help of the slope and y-intercept, you can draw a straight line through these points, and this line represents the graph of the equation. When graphing multiple linear equations on the same grid, their intersection points, if any, become particularly interesting, as they represent the solution to a system of equations.

Finding Intersection Points

The intersection point(s) of two or more lines on a graph represent the solution(s) to a system of equations. It is the point where the x and y values satisfy all the equations in the system simultaneously.

For instance, if we graph the equations \(y = 2x + 6\) and \(y = -2x + 6\), their point of intersection would provide the values of x and y that solve both equations. To find this point accurately, we can look for the coordinates where both lines cross each other on the grid.

In the given example, the two lines intersect at the point (1, 4). This means that substituting x with 1 in both equations yields y as 4. Hence, the solution to the system of equations is the point (1, 4). Intersection points can be a single point for two lines, infinitely many points if the lines are coincident, or none if the lines are parallel and never meet.