Certainly! Let's solve the problem step-by-step.
Step 1: Write down the given equations.
We have the following system of linear equations:
1. [tex]\( 2y + 3x = 16 \)[/tex]
2. [tex]\( 7y = 2x + 6 \)[/tex]
Step 2: Solve the system of equations to find the intersection point.
First, we rewrite the second equation in a standard form to match the first equation:
[tex]\[ 7y - 2x = 6 \][/tex]
Now, we have:
[tex]\[ 2y + 3x = 16 \][/tex]
[tex]\[ 7y - 2x = 6 \][/tex]
Next, to isolate the variables, we can use the method of elimination or substitution. Here, we'll use the elimination method.
Multiply the first equation by 2 and the second equation by 3 to align the coefficients of [tex]\(x\)[/tex]:
[tex]\[ \begin{cases}
4y + 6x = 32 \\
21y - 6x = 18
\end{cases} \][/tex]
Add these two equations to eliminate [tex]\(x\)[/tex]:
[tex]\[ 4y + 6x + 21y - 6x = 32 + 18 \][/tex]
[tex]\[ 25y = 50 \][/tex]
[tex]\[ y = 2 \][/tex]
Substitute [tex]\(y = 2\)[/tex] back into the first original equation to find [tex]\(x\)[/tex]:
[tex]\[ 2(2) + 3x = 16 \][/tex]
[tex]\[ 4 + 3x = 16 \][/tex]
[tex]\[ 3x = 12 \][/tex]
[tex]\[ x = 4 \][/tex]
Hence, the intersection point of the two lines is [tex]\( (4, 2) \)[/tex].
Step 3: Find the equation of the line through the point [tex]\((4, 2)\)[/tex] with a gradient of [tex]\(-2\)[/tex].
The general form of the equation of a line with a gradient [tex]\(m\)[/tex] passing through a point [tex]\((x_1, y_1)\)[/tex] is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Substitute [tex]\( m = -2 \)[/tex], [tex]\( x_1 = 4 \)[/tex], and [tex]\( y_1 = 2 \)[/tex]:
[tex]\[ y - 2 = -2(x - 4) \][/tex]
Simplify the equation:
[tex]\[ y - 2 = -2x + 8 \][/tex]
[tex]\[ y = -2x + 10 \][/tex]
Therefore, the equation of the line that passes through the intersection point [tex]\((4, 2)\)[/tex] with a gradient of [tex]\(-2\)[/tex] is:
[tex]\[ y = -2x + 10 \][/tex]
