Questions: Solve the system by graphing. For systems that do not have one unique solution, also state the number of solutions and whether the system is inconsistent or the equations are dependent. x-y=-1 3y=4x

Solution Steps

To graph the system of equations, we first need to rewrite each equation in the slope-intercept form, \(y = mx + b\).

1. For the first equation: \[ x - y = -1 \] Solving for \(y\), we get: \[ y = x + 1 \]

2. For the second equation: \[ 3y = 4x \] Solving for \(y\), we get: \[ y = \frac{4}{3}x \]

Now, we will graph the two equations:

1. Equation 1: \(y = x + 1\)

   • This line has a slope of 1 and a y-intercept of 1.
   • It passes through the points \((0, 1)\) and \((1, 2)\).

2. Equation 2: \(y = \frac{4}{3}x\)

   • This line has a slope of \(\frac{4}{3}\) and a y-intercept of 0.
   • It passes through the points \((0, 0)\) and \((3, 4)\).

To find the solution to the system, we need to determine where the two lines intersect. We set the equations equal to each other:

\[ x + 1 = \frac{4}{3}x \]

Solving for \(x\):

\[ 3(x + 1) = 4x \] \[ 3x + 3 = 4x \] \[ 3 = 4x - 3x \] \[ 3 = x \]

Substitute \(x = 3\) back into one of the original equations to find \(y\):

\[ y = 3 + 1 = 4 \]

Thus, the intersection point is \((3, 4)\).

Final Answer