Hello there. To solve this question, we'll have to remember some properties about system of linear equations and how to find the equation of a line in slope-intercept form.
Given the equation of a line
[tex]ax+dy=c[/tex]We can solve for y in order to find it in slope-intercept form
[tex]y=mx+b[/tex]Where m is the slope and b is the y-intercept.
For the first equation, we get
[tex]7x+3y=5[/tex]Solving for y, we first subtract 7x on both sides of the equation
[tex]3y=-7x+5_{}[/tex]Divide both sides of the equation by a factor of 3
[tex]y=-\frac{7}{3}x+\frac{5}{3}[/tex]And we can spot the slope as m = -7/3 and the y-intercept = 5/3.
For the second equation, we have
[tex]4x+3y=6[/tex]Solving for y, subtract 4x on both sides of the equation
[tex]3y=-4x+6[/tex]Divide both sides by a factor of 6
[tex]y=-\frac{4}{3}x+2[/tex]The slope in this case is equal to -4/3 and the y-intercept is equal to 2.
Finally, to determine if the system has one, infinite or no solutions, we make:
[tex]\begin{cases}7x+3y=5 \\ 4x+3y=6\end{cases}[/tex]The first way to make sure is that if we can find a single ordered pair (x, y) that satisfies this system. Subtract the second equation from the first, such that
[tex]\begin{gathered} 7x+3y-(4x+3y)=5-6 \\ 7x+3y-4x-3y=-1 \\ 3x=-1 \end{gathered}[/tex]Divide both sides by a factor of 3
[tex]x=-\frac{1}{3}[/tex]Plugging this into any equation, we find the solution for y
[tex]\begin{gathered} 4\cdot\mleft(-\dfrac{1}{3}\mright)+3y=6 \\ -\frac{4}{3}+3y=6 \\ 3y=6+\frac{4}{3}=\frac{22}{3} \\ y=\frac{22}{9} \end{gathered}[/tex]Hence the ordered pair that is a solution of this system of equations is
[tex]\mleft(-\dfrac{1}{3},\frac{22}{9}\mright)[/tex]This is in fact the unique solution to this system, since the lines only cross this time at x = -1/3 and y = 22/9.
The other way to check is if the determinant of the coefficients matrix is different of zero:
[tex]\begin{bmatrix}{7} & {3} \\ {4} & {3}\end{bmatrix}[/tex]Taking this determinant, that is, subtracting the products between the main and secondary diagonals, we have
[tex]7\cdot3-4\cdot3=21-12=9[/tex]That means that this system has only one solution.
For a system of linear equations (in fact, two lines)
