Let's solve step-by-step to find the coordinates of the intersection point of the two lines given by the equations:
1. The first equation is:
[tex]\[ x - 2y = 7 \][/tex]
2. The second equation can be simplified before using:
[tex]\[ 8x - 4y = 56 \][/tex]
First, we'll simplify the second equation by dividing every term by 4:
[tex]\[ \frac{8x - 4y}{4} = \frac{56}{4} \][/tex]
which simplifies to:
[tex]\[ 2x - y = 14 \][/tex]
So, we now have the system of equations as:
[tex]\[ x - 2y = 7 \][/tex]
[tex]\[ 2x - y = 14 \][/tex]
To eliminate one of the variables, we can modify the first equation to make the coefficients of [tex]\( y \)[/tex] match. We'll multiply the first equation by 2:
[tex]\[ 2(x - 2y) = 2 \cdot 7 \][/tex]
which simplifies to:
[tex]\[ 2x - 4y = 14 \][/tex]
Now, let’s subtract the second equation from this new equation:
[tex]\[ (2x - 4y) - (2x - y) = 14 - 14 \][/tex]
[tex]\[ 2x - 4y - 2x + y = 0 \][/tex]
[tex]\[ -3y = 0 \][/tex]
[tex]\[ y = 0 \][/tex]
With [tex]\( y = 0 \)[/tex], we substitute back into the first original equation:
[tex]\[ x - 2(0) = 7 \][/tex]
[tex]\[ x = 7 \][/tex]
Therefore, the coordinates of the intersection point of the two lines are:
[tex]\[ (7, 0) \][/tex]
So, the coordinates for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] at the intersection are:
[tex]\[ \boxed{7} \][/tex]
[tex]\[ \boxed{0} \][/tex]
