Problem 30 - Algebra 1

Question

Graph and check to solve the linear system. $$ \begin{aligned} &-3 x+y=10\\\ &7 x+y=20 \end{aligned} $$

Step-by-Step Solution

Answer

The solution to the system of equations is (2,16).

Step 1: Rewrite as y=mx+b (Slope-Intercept Form)

Rewrite the equations in the form \(y=mx+b\) by isolating y. For the first equation, add 3x to both sides to get \(y = 3x + 10\). For the second equation, subtract 7x from both sides to get \(y = -7x + 20\).

Step 2: Graph the Equations

On the same set of axes, graph both equations. The line \(y=3x+10\) will have a slope of 3 and y-intercept of 10. The line \(y=-7x+20\) will have a slope of -7 and a y-intercept of 20.

Step 3: Find the Intersection Point

The intersection point of two lines is the solution for the system of equations. In this case, the lines intersect at the point (2,16).

Step 4: Check the Solution

Check that (2,16) is a solution by substituting these values into both original equations. The checks are as follows: For \(-3x + y = 10\), check if \(-3(2) +16 = 10\). It holds true. For \(7x +y = 20\), check if \(7(2) +16= 20\), which holds true as well. Therefore, (2,16) is indeed the solution.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is pivotal in graphing and solving for intersections. This form is expressed as \( y = mx + b \), where \( m \) stands for the slope of the line, and \( b \) represents the y-intercept - the point where the line crosses the y-axis.

Converting a linear equation to the slope-intercept form simplifies the process of graphing since it directly reveals the steepness of the line (slope) and the starting point on the y-axis (y-intercept). For the given equations \( -3x + y = 10 \) and \( 7x + y = 20 \), they were rewritten as \( y = 3x + 10 \) and \( y = -7x + 20 \) respectively. With these conversions, the process of plotting becomes almost intuitive as you first mark the y-intercept on the graph and then use the slope to determine the direction and steepness of the line.

Graphing Linear Equations

Graphing linear equations involves plotting straight lines on a coordinate plane. Here's a straightforward approach to graph any linear equation in slope-intercept form:

• Begin at the y-intercept (b) on the y-axis.
• From the y-intercept, use the slope (m) to determine the rise over run, which tells you how many units to go up or down (rise) and how many units to go left or right (run).

For example, the line \( y = 3x + 10 \) has a y-intercept of 10. Start at the point (0, 10) on the graph. The slope is 3, which means for each step right (run), you go up (rise) three units. For the line \( y = -7x + 20 \), you start at (0, 20) and for every unit you step right, you go down seven units, due to the negative slope. By defining these two lines on the graph, we can visually interpret the system of equations.

Intersection Point of Lines

The intersection point of lines is the core of solving a system of linear equations by graphing. It represents the set of coordinates that satisfy both equations simultaneously. In graphical terms, it's where the two lines meet.

After graphing the lines using the slope-intercept form, identify the point where they cross. For the equations provided, the lines intersect at (2,16). This coordinate pair is the solution to the system as it makes both equations true. It is always good practice to double-check by substituting these values back into the original equations to ensure that they satisfy both, as shown in the step-by-step solution. If both equations are valid with the intersection point, you've found the correct solution to the system.