Solve the system of equations by graphing.
[tex]\left\{\begin{array}{l} x+y=9 \\ x-y=3 \end{array}\right.[/tex]
Use the graphing tool to graph the system.
Answer (2)
Rewrite the equations in slope-intercept form: y = − x + 9 and y = x − 3 .
Graph the two lines. Find the point of intersection by setting the two equations equal: − x + 9 = x − 3 .
Solve for x and y to find the solution: x = 6 , y = 3 .
Explanation
Analyze the problem We are given the following system of equations: x + y = 9 x − y = 3 Our goal is to solve this system by graphing. This means we need to find the point where the two lines intersect.
Rewrite equations First, let's rewrite each equation in slope-intercept form ( y = m x + b ). For the first equation, x + y = 9 , we subtract x from both sides to get: y = − x + 9 For the second equation, x − y = 3 , we subtract x from both sides to get − y = − x + 3 . Then, we multiply both sides by − 1 to get: y = x − 3
Graph the lines Now we have two equations in slope-intercept form: y = − x + 9 y = x − 3 We can graph these two lines. The first line has a slope of − 1 and a y-intercept of 9 . The second line has a slope of 1 and a y-intercept of − 3 .
Find the intersection point By graphing the two lines, we can find the point of intersection. Alternatively, we can solve the system algebraically. Since we have y = − x + 9 and y = x − 3 , we can set the two expressions for y equal to each other: − x + 9 = x − 3 Add x to both sides: 9 = 2 x − 3 Add 3 to both sides: 12 = 2 x Divide by 2 :
x = 6 Now substitute x = 6 into either equation to find y . Using the first equation: y = − 6 + 9 = 3 So the point of intersection is ( 6 , 3 ) .
State the solution Therefore, the solution to the system of equations is x = 6 and y = 3 .
Examples
Imagine you're trying to meet a friend. You know they started 9 blocks north of you and are walking south (Equation 1: x + y = 9). You also know they are currently 3 blocks further north than you are (Equation 2: x - y = 3). Solving this system of equations helps you figure out exactly where you'll meet them (at the intersection point), ensuring you don't miss each other. This method is useful in various scenarios like coordinating movements, balancing resources, or planning meetings.
To solve the system of equations x + y = 9 and x − y = 3 , we rewrite them in slope-intercept form as y = − x + 9 and y = x − 3 . By graphing these lines, we find their intersection point is ( 6 , 3 ) , meaning the solution is x = 6 and y = 3 .
;
