Problem 36 - Algebra 2

Question

Solve each system by graphing. \(\left\\{\begin{array}{c}{2 x+y=7} \\ {x+y=-5}\end{array}\right.\)

Step-by-Step Solution

Answer

The point of intersection is at (12, -17), which is the solution to the system.

Step 1: Write the equations in slope-intercept form

To graph the lines, rewrite each equation in the form of y=mx+b, where m is the slope and b is the y-intercept. For the first equation, solve for y: y = -2x + 7. For the second equation, solve for y: y = -x - 5.

Step 2: Plot the y-intercepts of both lines

Take the y-intercept (b) of each equation and plot the points on the y-axis. The y-intercept of the first line is 7, so plot the point (0,7). The y-intercept of the second line is -5, so plot the point (0,-5).

Step 3: Use the slope to find another point for each line

For the first line, with a slope of -2, from the y-intercept (0,7), move down 2 units and right 1 unit to find another point (1,5). For the second line, with a slope of -1, from the y-intercept (0,-5), move down 1 unit and right 1 unit to find another point (1,-6).

Step 4: Draw the lines on the graph

Draw the first line through the points (0,7) and (1,5). Draw the second line through the points (0,-5) and (1,-6). Extend both lines across the graph.

Step 5: Identify the point of intersection

Look for the point where the two lines intersect. This point is the solution to the system of equations.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is essential for graphing linear equations. It is the equation of a line written as \( y = mx + b \), where \( m \) is the slope of the line, and \( b \) is the y-intercept. The slope, \( m \) , indicates the steepness of the line and the direction in which the line moves. For every one unit that we move horizontally to the right, we move vertically by \( m \) units. A positive slope means the line is inclined upwards, while a negative slope means it's going downwards.

When we have a linear equation, the first thing we do is rearrange it into this form, which makes it much easier to graph. For instance, taking the equation \( 2x + y = 7 \) and solving for y, we get \( y = -2x + 7 \). This is now in slope-intercept form, making it clear that the slope is \( -2 \) and the y-intercept is \( 7 \).

Y-Intercept

The y-intercept is the point where the graph of an equation crosses the y-axis. It is represented by the \( b \) in the slope-intercept form \( y = mx + b \). This value can be seen as the starting point on a graph where your line will begin if you're plotting from left to right.

To find the y-intercept from an equation, set the value of \( x \) to zero and solve for \( y \). For the linear equation \( y = -x - 5 \), when \( x \) is zero, \( y \) is \( -5 \), which means that the line will pass through the point \( (0, -5) \) on the y-axis. When graphing, you start by marking this point and then use the slope to determine the next points of the line.

Graphing Linear Equations

Graphing linear equations involves plotting points and drawing a straight line through them. After determining the slope-intercept form, follow these steps to graph a linear equation:

• Plot the y-intercept on the graph.
• Use the slope \( m \) to find another point. If the slope is negative, move down and to the right from the y-intercept; if positive, move up and to the right.
• Draw a straight line through the points, extending beyond them to cover the entire graph area.

It is important to plot at least two points so you can align your ruler to these and draw an accurate line. As an exercise improvement advice, remember to label your axes and scale them equally to ensure that the steepness of the lines is visually correct.

Point of Intersection

The point of intersection is where two lines on a graph cross each other. It represents the solution to a system of linear equations, as it satisfies both equations simultaneously.

To identify the point of intersection, look for the exact coordinates where both lines intersect on the graph. For the system with equations \( y = -2x + 7 \) and \( y = -x - 5 \) right after graphing them, the intersection represents the values of \( x \) and \( y \) that make both equations true. This common solution is what you're often looking for when solving systems of equations by graphing. It's a good practice to estimate the point of intersection as precisely as possible and then verify by substituting the coordinates back into the original equations.