Problem 53 - Intermediate Algebra

Question

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.) $$ \left\\{\begin{array}{l} 4 x-3 y=5 \\ y=-2 x \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system is the intersection point: \((-1, 2)\).

Step 1: Rearrange the first equation

To graph the equations, we need both of them in slope-intercept form (\(y = mx + b\)). Start with the equation \(4x - 3y = 5\). To solve for \(y\), first isolate \(-3y\) by subtracting \(4x\) from both sides, resulting in \(-3y = -4x + 5\). Next, divide every term by \(-3\) to get \(y = \frac{4}{3}x - \frac{5}{3}\).

Step 2: Graph the equations

Now graph the two equations \(y = \frac{4}{3}x - \frac{5}{3}\) and \(y = -2x\) on a coordinate plane. The first equation has a y-intercept at \(-\frac{5}{3}\) and a slope of \(\frac{4}{3}\), meaning for every 3 units right, it goes up 4 units. The second equation has a y-intercept at 0 and a slope of -2, meaning it goes down 2 units for every 1 unit right.

Step 3: Identify the point of intersection

By graphing both lines, look for the point where they intersect. The point of intersection of the lines is where they have the same \(x\) and \(y\) values. Upon graphing, they intersect at \(x = -1\) and \(y = 2\).

Step 4: Verify the solution

Check if the point \((-1, 2)\) satisfies both equations. Substitute \(x = -1\) into both equations: \(y = -2(-1) = 2\) and for the other equation, \(4(-1) - 3(2) = -4 - 6 = -10\), which should equal the translated form of the second equation when simplified. Thus, this technique confirms the solution.

Key Concepts

Graphing Linear Equations

Graphing linear equations involves drawing the lines that represent each equation on the same coordinate plane in order to visually solve systems of equations. To begin, it is essential to plot each equation precisely. This is typically accomplished by first transforming each equation into the slope-intercept form, which makes it easier to interpret. In our exercise, the equations are \(y = \frac{4}{3}x - \frac{5}{3}\) and \(y = -2x\).

These lines are then plotted on a graph, with the y-axis representing the dependent variable \(y\), and the x-axis representing the independent variable \(x\).

• Identify the y-intercept, the point where the line crosses the y-axis. For instance, \(y = \frac{4}{3}x - \frac{5}{3}\) crosses the y-axis at \(-\frac{5}{3}\).
• Locate points by using the slope, a measure of the line's steepness. From the y-intercept, apply the slope to find other points.
• Plot the second equation, \(y = -2x\), which has a y-intercept of 0 and goes down 2 units for every 1 unit it moves to the right.

By drawing both lines on the same coordinate plane, one can easily spot their point of intersection. This visual method gives a direct way to solve a system of equations by seeing where the graphs meet.

Slope-Intercept Form

The slope-intercept form of a linear equation, given by \(y = mx + b\), is a widely used formula. Here, "\(m\)" stands for the slope of the line, and "\(b\)" is the y-intercept, where the line crosses the y-axis. This form is convenient for graphing, as it clearly displays critical information about the line.

In the problem exercise, the line expressed as \(y = \frac{4}{3}x - \frac{5}{3}\) has:

• Slope \(m = \frac{4}{3}\): This ratio indicates that for every three unit increases along the x-axis, the line rises by four units.
• Y-intercept \(b = -\frac{5}{3}\): This point on the graph is where the line touches the y-axis, at \(y = -\frac{5}{3}\).

The second line, \(y = -2x\), shows:

• Slope \(m = -2\): This negative slope means the line falls two units for every one unit it moves to the right.
• Y-intercept \(b = 0\): Here, the line begins at the origin.

By transforming equations into slope-intercept form, we gain clarity about how they will behave on a graph, which simplifies the process of plotting and identifying key features like intersection points.

Point of Intersection

The point of intersection in a system of linear equations represents the solution to both equations. It is the coordinate point that the lines share on a graph, signifying that \(x\) and \(y\) have the same value in both equations. To determine this point graphically, one must draw both lines accurately and find their crossing point.

In our step-by-step exercise solution, the lines \(y = \frac{4}{3}x - \frac{5}{3}\) and \(y = -2x\) intersect at \((-1, 2)\). This point is where both equations hold true simultaneously. After identifying the intersection point, it’s wise to verify by substitution:

• In \(y = -2x\), substituting \(x = -1\) yields \(y = -2(-1) = 2\).
• Substitute \(x = -1\) and \(y = 2\) into the other equation: \(4(-1) - 3(2) = -10\).

The verification process ensures that the solution is indeed correct, as it satisfies both equations. The point of intersection, therefore, not only represents a common solution but also serves as a powerful verification method for checking the validity of graphically depicted linear systems.