Answer :
Certainly! Let's solve the system of equations to determine the solution:
[tex]\[ \begin{array}{l} y = -\frac{1}{2} x + 2.5 \\ y = 2 x - 5 \end{array} \][/tex]
To find the point where these two lines intersect, we need to set the equations equal to each other and solve for [tex]\( x \)[/tex].
1. Set the two equations equal to each other:
[tex]\[ -\frac{1}{2} x + 2.5 = 2 x - 5 \][/tex]
2. Combine like terms to solve for [tex]\( x \)[/tex]:
First, get all [tex]\( x \)[/tex]-terms on one side of the equation and all constants on the other side:
[tex]\[ -\frac{1}{2} x - 2 x = -5 - 2.5 \][/tex]
Simplify the left and right sides:
[tex]\[ -\frac{1}{2} x - 2 x = -\frac{1}{2} x - 2 x = -2.5 x \][/tex]
[tex]\[ -5 - 2.5 = -7.5 \][/tex]
So we have:
[tex]\[ -2.5 x = -7.5 \][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-7.5}{-2.5} = 3 \][/tex]
4. Substitute [tex]\( x = 3 \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]. Let's use the second equation:
[tex]\[ y = 2 x - 5 \][/tex]
[tex]\[ y = 2(3) - 5 \][/tex]
[tex]\[ y = 6 - 5 \][/tex]
[tex]\[ y = 1 \][/tex]
Therefore, the solution to the system of equations is [tex]\((x, y) = (3, 1)\)[/tex].
So, the correct solution is:
[tex]\((3, 1)\)[/tex]
[tex]\[ \begin{array}{l} y = -\frac{1}{2} x + 2.5 \\ y = 2 x - 5 \end{array} \][/tex]
To find the point where these two lines intersect, we need to set the equations equal to each other and solve for [tex]\( x \)[/tex].
1. Set the two equations equal to each other:
[tex]\[ -\frac{1}{2} x + 2.5 = 2 x - 5 \][/tex]
2. Combine like terms to solve for [tex]\( x \)[/tex]:
First, get all [tex]\( x \)[/tex]-terms on one side of the equation and all constants on the other side:
[tex]\[ -\frac{1}{2} x - 2 x = -5 - 2.5 \][/tex]
Simplify the left and right sides:
[tex]\[ -\frac{1}{2} x - 2 x = -\frac{1}{2} x - 2 x = -2.5 x \][/tex]
[tex]\[ -5 - 2.5 = -7.5 \][/tex]
So we have:
[tex]\[ -2.5 x = -7.5 \][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-7.5}{-2.5} = 3 \][/tex]
4. Substitute [tex]\( x = 3 \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]. Let's use the second equation:
[tex]\[ y = 2 x - 5 \][/tex]
[tex]\[ y = 2(3) - 5 \][/tex]
[tex]\[ y = 6 - 5 \][/tex]
[tex]\[ y = 1 \][/tex]
Therefore, the solution to the system of equations is [tex]\((x, y) = (3, 1)\)[/tex].
So, the correct solution is:
[tex]\((3, 1)\)[/tex]
