Graph the system of linear equations.
$-\frac{1}{2} y=\frac{1}{2} x+5$ and $y=2 x+2$.
The solution to the system is ( $\square$ , $\square$ ).
Answer (2)
Rewrite the first equation in slope-intercept form: y = − x − 10 .
The second equation is already in slope-intercept form: y = 2 x + 2 .
Solve the system of equations by setting the two equations equal to each other: − x − 10 = 2 x + 2 , which gives x = − 4 .
Substitute the value of x back into either equation to find y : y = 2 ( − 4 ) + 2 = − 6 . The solution is ( − 4 , − 6 ) .
Explanation
Understanding the Problem We are given a system of two linear equations: − 2 1 y = 2 1 x + 5 and y = 2 x + 2 . We need to find the solution to this system, which is the point ( x , y ) where the two lines intersect. We are also asked to graph the two lines.
Rewriting the Equations First, let's rewrite the first equation in slope-intercept form ( y = m x + b ) . Multiply both sides of the equation − 2 1 y = 2 1 x + 5 by − 2 to get y = − x − 10 . The second equation is already in slope-intercept form: y = 2 x + 2 .
Finding Points for Graphing Now, let's graph both lines on the coordinate plane. To graph each line, we can find two points on the line. For the first line, y = − x − 10 , we can find the y-intercept by setting x = 0 , which gives y = − 10 . We can find the x-intercept by setting y = 0 , which gives x = − 10 . So, the two points are ( 0 , − 10 ) and ( − 10 , 0 ) . For the second line, y = 2 x + 2 , we can find the y-intercept by setting x = 0 , which gives y = 2 . We can find the x-intercept by setting y = 0 , which gives x = − 1 . So, the two points are ( 0 , 2 ) and ( − 1 , 0 ) .
Solving for x To find the solution to the system of equations, we can set the two equations equal to each other: − x − 10 = 2 x + 2 . Now, let's solve for x : 3 x = − 12 , so x = − 4 .
Solving for y Substitute the value of x back into either equation to find y . Using the second equation: y = 2 ( − 4 ) + 2 = − 8 + 2 = − 6 . So, the solution to the system is ( − 4 , − 6 ) .
Verifying the Solution Let's verify the solution by substituting x = − 4 and y = − 6 into both original equations. For the first equation: − 2 1 ( − 6 ) = 2 1 ( − 4 ) + 5 , which simplifies to 3 = − 2 + 5 , which is true. For the second equation: − 6 = 2 ( − 4 ) + 2 , which simplifies to − 6 = − 8 + 2 , which is also true.
Final Answer The solution to the system of equations is ( − 4 , − 6 ) .
Examples
Systems of linear equations are used in various real-world applications, such as determining the break-even point for a business, calculating the optimal mix of ingredients in a recipe, or modeling supply and demand in economics. In this case, finding the intersection of two lines can help determine the point where two different pricing or production strategies yield the same outcome, allowing for informed decision-making.
The solution to the system of linear equations − 2 1 y = 2 1 x + 5 and y = 2 x + 2 is the point ( − 4 , − 6 ) . By rewriting the first equation and finding the intersection of the lines, we determine this point. The solution can be graphically represented as the intersection of two lines on a coordinate plane.
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