State the solution to the following systems of equations:
The equation y = 5 x + 3 has been drawn on the given graph. A second line \\y = mx + b intersects this line at the one point \left(0, 3\right).
State the value of b. Explain your answer.
State the value that m cannot be equal to. Explain your answer.
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 sketch the graphs on the same coordinate plane.
State the values of x and y which satisfy both equations.
Consider the following linear equations:
\begin{aligned} y &= 2 x + 2 \\ y &= - 2 x + 2 \end{aligned}
Determine the gradient and y-intercept of the line y = 2 x + 2.
Determine the x\text{ and }y-intercepts of the line y = - 2 x + 2.
Hence sketch the graphs on the same coordinate plane.
State the values of x and y which satisfy both equations.
Consider the following linear equations:
\begin{aligned} y = - 4 x - 1 \\ y = - 4 x + 2 \end{aligned}
Sketch the two lines on the same coordinate plane.
Comment on the simultaneous solution of the two equations.
Consider the following linear equations:
\begin{aligned} 4 x - 2 y = 2 \\ - 2 x + 4 y = 2 \end{aligned}
Sketch the two lines on the same coordinate plane.
State the values of x and y which satisfy both equations.
Solve the following systems of linear equations:
\begin{aligned} y &= x - 6 \\ y &= 5 x + 6 \end{aligned}
\begin{aligned} y &= 5x + 34 \\ y &= 3x + 18 \end{aligned}
\begin{aligned} q &= 5p + 23 \\ q &= 3p + 25 \end{aligned}
\begin{aligned} y &= 5x + 12 \\ y &= 28x + 72 \end{aligned}
\begin{aligned} y &= x + 8 \\ y &= - 7 x + 16 \end{aligned}
\begin{aligned} y &= -4x -17 \\ 3y &= 21x + 147 \end{aligned}
\begin{aligned} y &= 3x -18 \\ x + y &= -2 \end{aligned}
\begin{aligned} y &= -6x - 26 \\ x + y &= - 6 \end{aligned}
\begin{aligned} d &= 3c + - 18 \\ c - d &= 10 \end{aligned}
\begin{aligned} y &= 2.3 x - 17.62 \\ y &= - 8.5 x + 30.98 \end{aligned}
\begin{aligned} y &= -5x + 22 \\ 6x + y &= 26 \end{aligned}
\begin{aligned} y &= 6x - 2 \\ 3x - y &= 7 \end{aligned}
\begin{aligned} m &= 8n - 30 \\ 5n + 9m &= 38 \end{aligned}
\begin{aligned} y &= 5x - 8 \\ 2x -3y &= -15 \end{aligned}
\begin{aligned} y &= -2x - 1 \\ x +2y &= 13 \end{aligned}
\begin{aligned} 2 a - 5 b &= - 10 \\ b &= - 5 a + 6 \end{aligned}
\begin{aligned} 4 x + 3 y &= 52 \\ 7 x - 5 y &= 9 \end{aligned}
Find the coordinates of the point that the two lines y = 3 x and y = - 4 x have in common.
Consider the following linear equations:
\begin{aligned} -8x -y =0 \\ -5x + 3y = 6 \end{aligned}
To solve this system using the elimination method, first multiply both sides of the first equation by 3 and then add the resulting equation to the second equation.
Write the new equation formed from the above procedure.
Find the x and y values that satisfy both equations.
Consider the following linear equations:
\begin{aligned} \dfrac{4x}{5} + \dfrac{3y}{5} = 4 \\ 8x - 3y = 4 \end{aligned}
State the operation required to change the fractional coefficients to integer coefficients in the first equation.
State the values of x and y that satisfy both equations.
Consider the following linear equations:
\begin{aligned} 3x +7y &=-6 \\ 2x - y &= -\dfrac{45}{7} \end{aligned}
What number should the second equation be multiplied by to eliminate y?
Eliminate y and hence find the value of x that satisfies both equations.
Substitute the x value to find the value of y that satisfies both equations.
Explain how to eliminate the variable y in the following systems of equations:
\begin{aligned} 2x + 3y &= 4 \\ 5x - y &= 3 \end{aligned}
\begin{aligned} 4x - 9y &= 8 \\ 5x + 7y &= 3 \end{aligned}
\begin{aligned} \dfrac{1}{7} x + \dfrac{4}{7} y = 7 \\ \dfrac{1}{2} x + \dfrac{2}{3} y = 5 \end{aligned}
\begin{aligned} 2 x - 0.7 y &= 5.9 \\ 0.65 x + 0.07 y &= - 0.02 \end{aligned}
For each of the following systems of linear equations:
Simplify and rewrite both equations so that all coefficients are integers.
Find the values of x and y that satisfy the equations.
\begin{aligned} \dfrac{7 x}{4} + \dfrac{y}{4} &= - \dfrac{1}{2} \\ \dfrac{x}{5} - \dfrac{9 y}{5} &= \dfrac{38}{7} \end{aligned}
\begin{aligned}-7p + 2q &= - \dfrac{13}{10} \\ -21p+10q &= - \dfrac{9}{10} \end{aligned}
\begin{aligned} - \dfrac{7 x}{9} + 3 y &= \dfrac{167}{45} \\ \dfrac{5 x}{4} + \dfrac{5 y}{4} &= - \dfrac{19}{4} \end{aligned}
For each of the following systems of linear equations:
Solve for the values of x and y that satisfy the equations.
Describe the system of equations as dependent, consistent or inconsistent.
\begin{aligned} 4x + 7y &= - 9 \\ 12 x + 21y &= 18 \end{aligned}
\begin{aligned} 3x -7y &= 4 \\ -12x + 28 y &= -16 \end{aligned}
Consider the following linear equations:
\begin{aligned} \dfrac{2x}{5} + \dfrac{3y}{5} &= -\dfrac{7}{5} \\ -\dfrac{1}{4} \left(-5x + \dfrac{5y}{9} \right) &= \dfrac{5}{9} \end{aligned}
Write the new equations after the fractional coefficients have been changed to integer coefficients and simplified.
Eliminate y and hence find the value of x that satisfies both equations.
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 integer coefficients, keeping the integers as small as possible.
Solve the new system for y by eliminating x.
Solve for x.
Solve the following simultaneous equations:
\begin{aligned} 2 x + 5 y &= 9 \\ x + y &= 10 \end{aligned}and hence find the value of:
x + 4 y
7 x + 10 y
3 y
Solve the following systems of equations:
\begin{aligned} 2 x + 5 y &= 44 \\ 6 x - 5 y &= - 28 \end{aligned}
\begin{aligned} 8 x + 3 y &= - 11 \\ - 8 x - 5 y &= 29 \end{aligned}
\begin{aligned}2 x - 5 y &= 1 \\ - 3 x - 5 y &= - 39 \end{aligned}
\begin{aligned}7 x - 4 y &= 15 \\ 7 x + 5 y &= 60 \end{aligned}
\begin{aligned} - 6 x - 2 y &= 46 \\ - 30 x - 6 y &= 246 \end{aligned}
\begin{aligned}- 5 x + 16 y &= 82 \\ 25 x - 4 y &= 122 \end{aligned}
\begin{aligned} \dfrac{x}{2} + y &= 3 \\ \dfrac{x}{5} + 3 y &= - 4 \end{aligned}
\begin{aligned}x + \dfrac{5}{4} y &= \dfrac{9}{4} \\ \dfrac{3}{5} x + y &= \dfrac{7}{5} \end{aligned}
\begin{aligned} - \dfrac{x}{4} + \dfrac{y}{5} &= 8 \\ \dfrac{x}{5} + \frac{y}{3} &= 1 \end{aligned}
\begin{aligned} 0.4 x - 0.63 y &= 0.23 \\ 2 x + 7 y &= - 9 \end{aligned}
\begin{aligned} 5 x + 3 y &= 7 \\ x + y &= 2 \end{aligned}
\begin{aligned} - 5 p - 7 q &= - \dfrac{43}{5} \\ -18p - 28q &= - \dfrac{187}{5} \end{aligned}
When we solve a system of equation using the elimination method, how do we know when there is no solution?
For each of the following systems of linear equations:
Find the number of solutions that satisfy both equations.
State whether the system of equations is dependent, consistent or inconsistent.
\begin{aligned}7 x + 3 y &= 4 \\ 14 x + 6 y &= 8 \end{aligned}
\begin{aligned}7 x + 3 y &= 5 \\ 28 x + 12 y &= 30 \end{aligned}
Consider the following system of linear equations:
\begin{aligned} - 6 x - 2 y &= - 28 \\ 2 x + 16 y &= 40 \\ 4 x - 2 y &= 12 \end{aligned}Find the values of x and y that satisfy the first two equations.
Determine if this solution satisfies the third equation by substituting the values of x and y into the left hand side of the equation.
Hence state whether the lines are concurrent.
The percentage of the workforce (y) that are teenagers is modelled by 3.3 x + y = 35.3. The percentage of the workforce that are pensioners is modelled by 3.2 x - y = - 28.8, where x is the number of years since 2018.
Use the elimination method to find the x and y values that satisfy both equations.
State the year in which the proportion of the workforce that are teenagers and the proportion of the workforce that are pensioners is the same.
State the percentage of the workforce that are teenagers (or the percentage of the workforce that are pensioners) in this year.
Consider the straight line y = a x + b that passes through the two points \left(5, 3\right) and \left(8, 0\right).
Write a pair of simultaneous equations using the points given.
Find the value of a and b.
State the equation of the straight line that passes through the points \left(5, 3\right) and \left(8, 0\right).