# StudySquare

# AQA A-level Maths Solving equations graphically

This page covers the following topics:

A straight line graph has an equation of y = mx + c, where m is the gradient of the graph and c is the y-intercept of the graph, ie. the point at which the line crosses the y-axis. The gradient of the line can be calculated using: m = change in y/change in x. Another equation that can be used to find the equation of a straight line is y − y₁ = m(x − x₁), where m is the gradient of the graph and the graph passes through the point (x₁, y₁).

A quadratic equation is written in the form y = ax² + bx + c. To find the equation of a quadratic from its graph, the values of a, b and c must be calculated. To do so, three points on the graph must be known. These can be substituted in the general form of a quadratic, which would give 3 equations that can be solved simultaneously to find the values of a, b and c.

Systems of simultaneous equations can be solved by graphing: the solutions to the system will be the points of intersection on their graphs.

Graphs that involve negative quadratic equations and cubic equations can be solved just as linear and quadratic equations. The typical shapes of a negative quadratic and cubic graphs are given in the diagram.

# 1

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Find the equation of the straight line in the diagram, given that it passes through (0, 3) and (6, 5).

# 2

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The line y = 7x + 21 meets the x-axis at the point W. Find the equation of the line with a gradient m = −4 that passes through point W.

# 6

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The graphs of a quadratic and a linear equation are drawn on the same set of axes and they do not intersect. The linear equation is substituted into the quadratic equation to form another quadratic equation. What could we say about the discriminant of this quadratic equation? Explain your answer.

# 7

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The graphs of 2x + y = 6 and 5x − y = 1 are given on the same set of axes. Use the graph to determine the solution to this system of simultaneous equations.

# 8

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Graphically determine the number of solutions to the system of simultaneous equations: y + x = 8, y = x² + 12x + 20.

# 9

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The graphs of a quadratic equation and a cubic equation are drawn on the same set of axes. Given that the two graphs intersect twice, state the number of solutions for this pair of simultaneous equations.

# 10

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The graphs of y = 2x + 4 and y = x³ − 3x² − 4x + 12 are given on the same set of axes. Determine the solutions to this system of simultaneous equations.

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