Problem 27 - Algebra Form and Function

Question

Solve the system of equations graphically. $$ \left\\{\begin{array}{r} 2 x+5 y=7 \\ -3 x+2 y=1 \end{array}\right. $$

Step-by-Step Solution

Answer

Question: Solve the following system of equations graphically and verify the solution. $$ 2x + 5y = 7 \\ -3x + 2y = 1 $$ Answer: After graphing the two lines, determine the intersection point and verify it in the original equations to confirm the solution.

Step 1: Rewrite each equation in y = mx + b form

To rewrite the equations in the desired form, we will isolate y in both equations. $$ 2x + 5y = 7 \\ 5y = -2x + 7 \\ y = -\frac{2}{5}x + \frac{7}{5} $$ $$ -3x + 2y = 1 \\ 2y = 3x + 1 \\ y = \frac{3}{2}x + \frac{1}{2} $$ Now, we have the equations in y = mx + b form: $$ y = -\frac{2}{5}x + \frac{7}{5} \\ y = \frac{3}{2}x + \frac{1}{2} $$

Step 2: Graph the lines

Now, graph the lines using their respective slope and y-intercept. Line 1: Slope: \(-\frac{2}{5}\) Y-intercept: \(\frac{7}{5}\) Line 2: Slope: \(\frac{3}{2}\) Y-intercept: \(\frac{1}{2}\) Plot the y-intercepts, and use the slopes to plot additional points for each line. Then, draw the lines through these points.

Step 3: Determine the intersection point

Observe the graph and identify the point where both lines intersect. This intersection point represents the solution to the system of equations because it satisfies both equations simultaneously.

Step 4: Verify the intersection point in the original equations

To ensure the accuracy of the intersection point, plug its x and y values back into the original system of equations to check if both equations are satisfied. If both are true, the intersection point is correct, and the system is solved.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that describe a straight line on a graph. They are crucial in algebra because they represent the simplest form of relationships between two variables. A typical linear equation in two variables, such as \(x\) and \(y\), can be represented as \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. Here, \(x\) and \(y\) are the variables that we often solve for.

In practical terms, a linear equation depicts a relationship where the change in one variable is proportional to the change in another. This proportion is represented as a straight line when plotted on a coordinate plane. Linear equations can have one or two variables. In the context of two variables, they form lines like those we graphically solve for points of intersection, finding common solutions to systems of equations.

• Linear equations form straight lines on graphs.
• The general form is \(ax + by = c\).
• They reveal relationships with constant rates of change.

Slope-Intercept Form

The slope-intercept form of a linear equation is a way of expressing the equation that makes it easy to graph and understand. It is written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

The slope \(m\) tells us how steep the line is; it represents the rate of change or how much \(y\) changes with a unit change in \(x\). If \(m\) is positive, the line slopes upwards from left to right. If \(m\) is negative, the line slopes downwards.

The y-intercept \(b\) is the point where the line crosses the y-axis, meaning it's the value of \(y\) when \(x\) is zero. Identifying the slope and y-intercept from this form allows us to quickly plot the line on a graph without further calculation.

• Slope-intercept form: \(y = mx + b\).
• \(m\) is the slope, indicating the line's steepness and direction.
• \(b\) is the y-intercept, showing where the line meets the y-axis.

Intersection Point

The intersection point of two lines representing a system of linear equations is a key concept in algebra because it provides the solution to the system. This point is where both equations are satisfied simultaneously. In a graph, it is the point where the two lines cross.

To find the intersection point graphically, we use the slope-intercept form to draw both lines on the same coordinate plane. The point where they intersect is the solution to the system, giving the values of \(x\) and \(y\) that satisfy both equations.

Finding this point involves identifying the precise coordinates on the graph where the lines meet. In practical exercises, once the intersection point is found, it should be verified by plugging its coordinates back into the original equations to ensure they hold true. This verification ensures accuracy and confirms the solution.

• Intersection points provide the solution to systems of equations.
• They are found where two lines cross on a graph.
• Verification by substitution ensures the solution's correctness.