1.02c Simultaneous equations: two variables by elimination and substitution

In this question you must show all stages of your working. Give your answers in fully simplified surd form. Given that \(a\) and \(b\) are positive constants, solve the simultaneous equations \begin{align} a - b &= 8
\log_5 a + \log_5 b &= 3 \end{align} [6]

Given that \(a\) and \(b\) are positive constants, solve the simultaneous equations \(a = 3b\), \(\log_3 a + \log_3 b = 2\). Give your answers as exact numbers. [6]

Given that \(a\) and \(b\) are positive constants, solve the simultaneous equations $$a = 3b,$$ $$\log_3 a + \log_3 b = 2.$$ Give your answers as exact numbers. [6]

Solve the simultaneous equations $$x - 2y - 1 = 0$$ $$x^2 + 4y^2 - 10x + 9 = 0$$ [7]

Solve the simultaneous equations $$x - 3y + 1 = 0,$$ $$x^2 - 3xy + y^2 = 11.$$ [7]

\includegraphics{figure_1} Figure 1 shows the curve with equation \(y^2 = 4(x - 2)\) and the line with equation \(2x - 3y = 12\). The curve crosses the \(x\)-axis at the point \(A\), and the line intersects the curve at the points \(P\) and \(Q\).

1. Write down the coordinates of \(A\). [1]
2. Find, using algebra, the coordinates of \(P\) and \(Q\). [6]
3. Show that \(\angle PAQ\) is a right angle. [4]

The straight line \(l_1\) has equation \(4y + x = 0\). The straight line \(l_2\) has equation \(y = 2x - 3\).

1. On the same axes, sketch the graphs of \(l_1\) and \(l_2\). Show clearly the coordinates of all points at which the graphs meet the coordinate axes. [3]

The lines \(l_1\) and \(l_2\) intersect at the point \(A\).

1. Calculate, as exact fractions, the coordinates of \(A\). [3]
2. Find an equation of the line through \(A\) which is perpendicular to \(l_1\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [3]

Solve the simultaneous equations \(x - 3y + 1 = 0\), \(x^2 - 3xy + y^2 = 11\). [7]

A circle has equation \(x^2 + y^2 = 45\).

1. State the centre and radius of this circle. [2]
2. The circle intersects the line with equation \(x + y = 3\) at two points, A and B. Find algebraically the coordinates of A and B. Show that the distance AB is \(\sqrt{162}\). [8]

Find the coordinates of the point of intersection of the lines \(y = 3x + 1\) and \(x + 3y = 6\). [3]

Find the coordinates of the points of intersection of the circle \(x^2 + y^2 = 25\) and the line \(y = 3x\). Give your answers in surd form. [5]

\includegraphics{figure_11} Fig. 11 shows the line joining the points A \((0, 3)\) and B \((6, 1)\).

1. Find the equation of the line perpendicular to AB that passes through the origin, O. [2]
2. Find the coordinates of the point where this perpendicular meets AB. [4]
3. Show that the perpendicular distance of AB from the origin is \(\frac{9\sqrt{10}}{10}\). [2]
4. Find the length of AB, expressing your answer in the form \(a\sqrt{10}\). [2]
5. Find the area of triangle OAB. [2]

Find the coordinates of the point of intersection of the lines \(y = 3x - 2\) and \(x + 3y = 1\). [4]

Solve the simultaneous equations \begin{align} x + y &= 2
3x^2 - 2x + y^2 &= 2 \end{align} [7]

The straight line \(l_1\) has the equation \(3x - y = 0\). The straight line \(l_2\) has the equation \(x + 2y - 4 = 0\).

1. Sketch \(l_1\) and \(l_2\) on the same diagram, showing the coordinates of any points where each line meets the coordinate axes. [3]
2. Find, as exact fractions, the coordinates of the point where \(l_1\) and \(l_2\) intersect. [3]

Find the pairs of values \((x, y)\) which satisfy the simultaneous equations $$3x^2 + y^2 = 21$$ $$5x + y = 7$$ [7]

Solve the simultaneous equations \begin{align} x - 3y + 7 &= 0
x^2 + 2xy - y^2 &= 7 \end{align} [7]

The straight line \(l_1\) has gradient \(\frac{3}{4}\) and passes through the point \(A (5, 3)\).

1. Find an equation for \(l_1\) in the form \(y = mx + c\). [2]

The straight line \(l_2\) has the equation \(3x - 4y + 3 = 0\) and intersects \(l_1\) at the point \(B\).

1. Find the coordinates of \(B\). [3]
2. Find the coordinates of the mid-point of \(AB\). [2]
3. Show that the straight line parallel to \(l_2\) which passes through the mid-point of \(AB\) also passes through the origin. [4]