The following graph displays a system of two equations:
State the solution to the system in the form \left(x, y\right).
The following graph displays a system of two linear equations:
Determine the number solutions for the system.
Consider the following linear equations:
\begin{aligned} y &= \dfrac{x}{3} + \dfrac{1}{3} \\ - 8 y &= 8 x + 8 \end{aligned}
Determine the x\text{ and }y-intercepts of the line y = \dfrac{x}{3} + \dfrac{1}{3}.
Determine the x\text{ and }y-intercepts of the line - 8 y = 8 x + 8.
Hence plot the graphs of the 2 equations on the same coordinate plane.
State the values of x and y which satisfy both equations.
For each of the following pairs of equations:
Sketch the graphs of the two equations on the same coordinate plane.
State the values of x and y which satisfy both equations.
\begin{aligned} 4 x - 2 y &= 2 \\ - 2 x + 4 y &= 2 \end{aligned}
\begin{aligned} y &= 5 x - 7 \\ y &= - x + 5 \end{aligned}
x + y = 7 \\ x - y = 3
\begin{aligned} x + y &= 7 \\ 4x + 5y &= 32 \end{aligned}
\begin{aligned} y &= \dfrac{x}{2} - 3 \\ y &= 6 - x \end{aligned}
\begin{aligned} y &= \dfrac{3}{2}x + 4 \\ y &= \dfrac{-1}{2} x + 8 \end{aligned}
Solve the following systems of equations using the substitution method:
\begin{aligned} y &= 5x + 34 \\ y &= 3x + 18 \end{aligned}
\begin{aligned} y &= -4x - 17 \\ 3y &= 21x + 147 \end{aligned}
\begin{aligned} x &= 4 y - 27 \\ 2 y + x &= 21 \end{aligned}
\begin{aligned} x &= 3 y - 21 \\ 8 y + 5 x &= 79 \end{aligned}
\begin{aligned} y &= 3 x - 18 \\ x - y &= 10 \end{aligned}
\begin{aligned} y &= -5 x + 22 \\ 6 x + y &= 26\end{aligned}
Solve the following pairs of simultaneous equations using the elimination method:
Solve the following pairs of simultaneous equations using an appropriate algebraic method:
\begin{aligned} x + y &= 8 \\ 2x - 3y &= 26 \end{aligned}
\begin{aligned} y &= 2x + 16 \\ y &= 3x + 21 \end{aligned}
\begin{aligned} y &= 4x - 17 \\ x + y &= 38 \end{aligned}
\begin{aligned} 3x + y &= 22 \\ y &= -2x + 10 \end{aligned}
\begin{aligned} 2x + y &= 5 \\ 5x + 3y &= 9 \end{aligned}
\begin{aligned} 11x + 6y &= 27 \\ 7x + 6y &= -9 \end{aligned}
\begin{aligned} 6x + 10y &= 59 \\ -4x + 5y &= 19 \end{aligned}
\begin{aligned} 18x - 7y &= -64 \\ 3x + 5y &= 14 \end{aligned}
\begin{aligned} 5x - 3y &= 46 \\ 3x + 10y &= 4 \end{aligned}
\begin{aligned} \dfrac{x}{2} + 3y &= -6 \\ - \dfrac{x}{4} + y &= -7 \end{aligned}
Consider the following linear equations:
\begin{aligned} 0.2 x + 0.3 y &= 0.5 \\ 0.5 x + 0.4 y &= 0.2 \end{aligned}
Rewrite the system of equations as an equivalent system with the smallest possible integer coefficients.
Solve the new system for x and y.
Consider the straight line y = m x + c that passes through the two points \left(2, 0\right) and \left(6, - 5 \right).
Write a pair of simultaneous equations using the points given.
Solve the equations for m and c.
Hence state the equation of the straight line that passes through the two points.
Consider the following system of equations:
\begin{aligned} x^{2} &= y +14 \\ y &= 3x - 16 \end{aligned}
The graph suggests that the two points of intersection are \left(1, - 13 \right) and \left(2, - 10 \right).
Verify that these are the points of intersection by substituting into the equation.
The following functions are displayed on the graph below:
\begin{aligned} y &= x^{2} \\ y &= - 3 x \end{aligned}State the coordinates of the points of intersection.
Hence state the solutions to the equation x^{2} = - 3 x.
Consider the quadratic function y = x^{2} + 4 and the linear function x + y = 4.
Sketch the graphs of y = x^{2} + 4 and x + y = 4 on the same coordinate plane.
Hence determine the number of solutions to the equation x^{2} + 4 = -x + 4.
Find the points of intersection of the two functions.
For each of the following pairs of equations:
Sketch the graphs of the two equations on the same coordinate plane.
State the coordinates of the points of intersection.
\begin{aligned} y &= x^{2} \\ y &= 2 x^{2} - 1 \end{aligned}
\begin{aligned} y &= x^{2} + 1 \\ y &= 3 - x^{2} \end{aligned}
The function y = x^{2} + 1 is shown on the graph:
Determine the number of points of intersection the line y = -2 has with the parabola.
State the minimum value of k such that the line y = k has at least one point of intersection with the parabola.
Vanessa is using algebra to find the points of intersection of a parabola and a straight line. She finds a quadratic equation in the form a x^{2} + b x + c = 0, but when she tries to solve it, she finds that the equation has no solutions.
State whether following graphs could represent the parabola and straight line in this situation.
Find the coordinates of the points where the vertical line x = - 5 intersects the curve \\ y = - 2 x^{2} + x - 12.
Find the coordinates of the points of intersection for the following systems of equations:
\begin{aligned} y&= x^{2} \\ y&=9 \end{aligned}
\begin{aligned} y&=x^{2}-8x \\ y&=-7 \end{aligned}
One point of intersection of the curves y = x^{2} - 2 x + 30 and y = k x + 10 occurs at x = 4. Find the value of k.
Consider the following pair of parabolas:
\begin{aligned} y &= 2 x^{2} + 11 x + 1 \\ y &= x^{2} + 3 x - 3 \end{aligned}Find the exact value of the x-coordinates of their points of intersection.
The function y = - x^{2} - 2 x + 3 is shown on the graph:
State the linear expression that must be added to -x^{2} + x + 2 to form \\ -x^{2} - 2x + 3
Hence state the equation of the straight line that would need to be graphed to solve the equation \\ -x^{2} + x + 2 = 0 .
Graph the straight line on the same coordinate plane.
Hence determine the solutions to \\\ -x^{2} + x + 2 = 0.