Question

Graph and check to solve the linear system. $$ \begin{aligned} &0.5 x+0.6 y=5.4\\\ &-x+y=9 \end{aligned} $$

Short answer

The linear system has the solution at the point (0.4, 9.4).

Step-by-step solution

Convert equations to slope-intercept form

For the first equation, divide both sides by 0.6 to isolate y: \(y = -\frac{5}{6}x + 9\). For the second equation, add x to both sides to isolate y: \(y = x + 9\).

Graph the equations

Plot both lines on the same set of axes. The line from the first equation has a slope of \(-\frac{5}{6}\) and y-intercept of 9. The line from the second equation has a slope of 1 and y-intercept of 9.

Find the point of intersection

Observe where the two lines intersect. This point of intersection is the solution to the system of equations. In this case, both lines intersect at \(x = 0.4, y = 9.4\).

Key concepts

Slope-Intercept Form

The slope-intercept form is one of the most convenient ways to represent a linear equation. It has the format \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept of the line. The slope is the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line, and the y-intercept is the point where the line crosses the y-axis.

In the given exercise, converting the equations to the slope-intercept form is crucial for graphing. For example, the first equation is transformed into \( y = -\frac{5}{6}x + 9 \) by dividing both sides by 0.6. This tells us that for every 6 units we move horizontally, the vertical position of the line changes by -5 units, and the line crosses the y-axis at 9. Simplifying the slope-intercept form makes graphing more straightforward and allows for easier identification of the lines' behaviors on a graph.

Graphing Linear Equations

To visualize and solve a system by graphing, we take each linear equation in slope-intercept form and plot its corresponding line on a graph. Key elements like the slope and y-intercept guide us in drawing the lines accurately.

Creating the Graph:

Start by plotting the y-intercept on the y-axis. From there, use the slope to determine the direction and steepness of the line. In our exercise, the first line begins at a y-intercept of 9 and slopes downwards since the slope is negative. For each step rightward (positive x-direction), the line descends by 5/6 of a unit. Conversely, the second line also starts at a y-intercept of 9 but, with a slope of 1, it ascends at a 45-degree angle, indicating equal horizontal and vertical movement. By drawing both lines on the same axes, we can examine how they relate to one another.

Point of Intersection

The point of intersection is where two or more lines on a graph cross each other. This point is especially significant when solving linear systems as it represents the solution set — the values of x and y that satisfy all equations in the system.

To determine the point of intersection, observe where the lines plotted converge. In our example, after graphing each line according to their slope and y-intercept, we see that they intersect at the coordinates (0.4, 9.4). This point means that an x-value of 0.4 and a y-value of 9.4 solve both original equations simultaneously. Verifying by substituting these values back into the original equations is a good practice to ensure the solution is correct. When graphing, precision is important, and technology, such as graphing calculators or software, can be quite helpful in identifying the exact point of intersection.