Problem 22 - Algebra 1

Question

Graph and check to solve the linear system. $$ \begin{aligned} &y=2 x-4\\\ &y=-\frac{1}{2} x+1 \end{aligned} $$

Step-by-Step Solution

Answer

The lines intersect at the point (1, -2), so the solution to the system of equations is \(x=1\), \(y=-2\)

Step 1: Graph the First Equation

Start by graphing the equation \(y=2x-4\). This is a straight line with a slope of 2 and a y-intercept of -4. The line crosses the y-axis at the point (0, -4) and rises 2 units for every 1 unit it goes to the right.

Step 2: Graph the Second Equation

Next, graph the equation \(y=-{1}/{2}x +1\). This is also a straight line, but with a slope of -1/2 and a y-intercept of 1. The line crosses the y-axis at the point (0,1) and drops 1 unit for every 2 units it goes to the right.

Step 3: Find the Point of Intersection

The point of intersection of the two lines is the solution to the system of equations. By observing the two graphs together, it can be seen that they intersect at the point (1, -2). Thus, the solution to the system of equations is \(x=1\), \(y=-2\)

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to express linear equations, allowing us to graph them easily. It has the general format of \(y = mx + b\), where:

• \(m\) is the slope, representing the rate of change of the line. This tells us how steep the line is.
• \(b\) is the y-intercept, indicating where the line crosses the y-axis.

Thus, for the equation \(y = 2x - 4\), the slope \(m\) is 2 and the y-intercept \(b\) is -4. This means the line rises 2 units vertically for every 1 unit it moves horizontally. For the second equation, \(y = -\frac{1}{2}x + 1\), the slope is \(-\frac{1}{2}\) and the y-intercept is 1. The negative slope indicates the line decreases 1 unit for every 2 units it moves horizontally to the right. Understanding this form is crucial for graphing linear equations efficiently.

Graphing Linear Equations

Graphing linear equations is an essential skill that allows for visualization of solutions. Start with each line's y-intercept, the point \(b\) on the y-axis. For \(y = 2x - 4\), plot the point (0, -4) on the y-axis. This is where the line starts.

• Use the slope \(2\) to find the next point. Since the slope is 2, move up 2 units and 1 unit to the right, landing at (1, -2). Continue this pattern to draw the line.

For \(y = -\frac{1}{2}x + 1\), begin at (0, 1) on the y-axis.

• With a slope of \(-\frac{1}{2}\), move 1 unit down and 2 to the right for the next point, reaching (2, 0). Repeat this pattern to complete the line.

It's critical to extend the lines sufficiently to see where they might intersect. The point of intersection is meaningful as it provides the solution to the system of linear equations.

System of Equations

A system of equations consists of multiple equations that share the same variables. Solving these systems involves finding the values that satisfy all equations simultaneously. The solution to a system can be visualized as the point where all graphs intersect. In our example, the intersection point represents a common solution to both linear equations. By analyzing the graph, observe the intersection at the point (1, -2). This means that \(x = 1\) and \(y = -2\) satisfy both equations \(y = 2x - 4\) and \(y = -\frac{1}{2}x + 1\).

• Solve systems by graphing each line and identifying their intersection.
• Alternatively, use algebraic methods like substitution or elimination for more precision.

Understanding systems of equations allows us to solve practical problems where multiple conditions or constraints interact.