Problem 5 - Algebra 1

Question

Graph the linear system below. Then decide if the ordered pair is a solution of the system. $$ \begin{array}{l} -x+y=-2 \\ 2 x+y=10 \end{array} $$ $$ (-4,2) $$

Step-by-Step Solution

Answer

The point (-4,2) is a solution to the system if and only if it lies on both the lines of the graph.

Step 1: Rewrite the equations in slope-intercept form

The first equation can be rewritten as \(y = x - 2\), and the second equation can be rewritten as \(y = 10 - 2x\). Now the equations are in slope-intercept form \(y = mx + b\), which is easier to graph.

Step 2: Graph the lines

Begin by plotting the y-intercepts of both equations, which are -2 and 10 respectively, then use the slope of each equation to plot a second point for each line. Draw a line through these points to represent each equation. The first line should slope upwards and the second should slope downwards.

Step 3: Check the point

The point (-4,2) can be found by plotting -4 on the x-axis and 2 on the y-axis. If this point lies on both lines, then it is a solution to the system of equations.

Step 4: Conclusion

By analyzing where the point (-4,2) falls relative to the two lines drawn, it can be determined whether or not it is a solution to the system of equations. If it falls on both lines, then it is a solution. If it falls on only one or none of the lines, then it is not a solution to the system.

Key Concepts

Graphing

Graphing is a way to visually represent equations on a coordinate plane. To graph a linear equation, you'll typically start by identifying key elements such as the y-intercept and slope. These will help you draw the line accurately.

• Identify the y-intercept: The y-intercept is where the line crosses the y-axis. It tells you where to place your first point.
• Use the slope: The slope indicates how steep the line is. A positive slope means the line rises, while a negative slope means it falls.
• Plot multiple points: Using the y-intercept and slope, calculate and plot a second point. Connect these points with a straight line.

Graphing allows you to see the solution visually. When graphing a system of equations, the solution is located where the lines intersect. This makes it a powerful tool for solving equations.

Solution of a System

The solution of a system of equations is the set of values that make all equations in the system true simultaneously. Typically, in a system of linear equations, you're looking for where two lines intersect on the graph.

• Intersection point: This point is the set of \((x, y)\) coordinates that satisfy both equations.
• Unique solutions: If the lines intersect at a single point, that \((x, y)\) is the only solution.
• No solution: If the lines are parallel and never intersect, the system has no solution.
• Infinite solutions: If the lines overlap, every point on the line is a solution.

In our exercise, the point \((-4, 2)\) is checked to see if it lies on both lines, indicating if it is a solution to the system.

Slope-Intercept Form

The slope-intercept form is a way of writing linear equations as \(y = mx + b\). In this format, the slope and y-intercept are easily identifiable:

• Slope (m): Indicates the steepness of the line. A larger value means a steeper slope.
• Y-intercept (b): The point where the line crosses the y-axis at \((0, b)\).

Putting an equation into slope-intercept form makes it straightforward to graph because:

• You can immediately plot the y-intercept.
• You can use the slope to find another point.

Using slope-intercept form simplifies understanding the graph's shape and direction, making it much easier to visualize and solve linear systems.